ENGIN. 
LIBRARY 


UC-NRLF 


B    3    HE 


Mechanics  Department 


Engineering 
Library 


'« 


FORMULAS 


IN 


GEAR  ING 


THIRD  EDITION. 


WITH    PRACTICAL    SUGGESTIONS. 


PROVIDENCE,  R.  I. 

BROWN    &    SHARPE     MANUFACTURING    COMPANY 

1900. 


Engineering- 
Library 


,    • 


":/;  V  ': 


DEPT. 


Entered  according  to  Act  of  Congress,  in  the  year  1900  by 

BROWN  &  SHARPE  MF(r.  CO., 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 

Registered  at  Stationers'  Hall,  London,  Eng. 

All  rights  reserved. 


.PREFACE. 

It  is  the  aim,  in  the  following  pages,  to  condense  as  much 
as  possible  the  solution  of  all  problems  in  gearing  which  in  the 
ordinary  practice  may  be  met  with,  to  the  exclusion  of  prob- 
lems dealing  with  transmission  of  power  and  strength  of 
gearing.  The  simplest  and  briefest  being  the  symbolical 
expression,  it  has,  whenever  available,  been  resorted  to.  The 
mathematics  employed  are  of  a  simple  kind,  and  will  present 
no  difficulty  to  anyone  familiar  with  ordinary  Algebra  and 
the  elements  of  Trigonometry. 


735767 


CONTENTS. 

FORMULAS    IN    GEARING. 


CHAPTER  I. 

PAGB 
Systems  of  Gearing ...  .  .       i 

CHAPTER    II. 

Spur  Gearing— Formulas — Table  of  Tooth   Parts — Comparative  Sizes 

of  Gear  Teeth 4 

CHAPTER  III. 

Bevel  Gears,  Axes  at  Right  Angles — Formulas — Bevel  Gears,  Axes  at 
any  Angle — Formulas — Undercut  in  Bevel  Gears — Diameter  Incre- 
ment— Tables  for  Angles  of  Edge  and  Angles  of  Face — Tables  of 
Natural  Lines 1 1 

CHAPTER  IV. 

Worm  and  Worm  Wheel,  Formulas — Undercut  in  Worm  Wheels — 

Table  for  gashing-  Worm  Wheels 34 

CHAPTER  V. 

Spiral  or  Screw  Gearing — Axes  Parallel — Axes  at  Right  Angles — 
Axes  at  any  Angle — General  Formulas— Table  of  Prime  Num- 
bers and  Factors 4° 

CHAPTER  VI. 
Internal  Gearing — Internal  Spur  Gearing — Internal  Bevel  Gears 58 

CHAPTER  VII. 
Gear  Patterns 64 

CHAPTER  VIII. 
Dimensions  and  Form  for  Bevel  Gear  Cutters 67 

CHAPTER  IX. 
Directions  for  cutting  Bevel  Gears  with  Rotary  Cutter 70 

CHAPTER    X. 
The  Indexing  of  any  Whole  or  Fractional  Number 73 

CHAPTER  XL 

The  Gearing  of  Lathes  for  Screw  Cutting — Simple  Gearing — Compound 

Gearing — Cutting  a  Multiple  Screw         77 


FORMULAS   IN    GEARING. 


I. 


SYSTEMS  OF  GEARING. 

(Figs,  i,  2.) 

There  are  in  common  use  two  systems  of  gearing,  viz.:  the 
involute  and  the  epicycloidal. 

In  the  involute  system  the-outlines  of  the  working  parts  of  a 
tooth  are  single  curves,  which  may  be  traced  by  a  point  in  a 
flexible,  inextensible  cord  being  unwound  from  a  circular  disk 
the  circumference  of  which  is  called  the  base  circle,  the  disk 
being  concentric  with  the  pitch  circle  of  the  gear. 


In  Fig.  i  the  two  base  circles  are  represented  as  tangent  to 
the  line  P  P.  This  line  (P  P)  is  variously  called  "  the  line  of 
pressure,"  '"  the  line  of  contact,"  or  "  the  line  of  action." 


BROWN    &    SHARPE    MFG.    CO. 


In  our  practice  this  is  drawn  so  as  to  make  with  a  normal 
to  the  center  line  (O  O')  14/4°,  or  with  the  center  line  75%°. 

The  rack  of  this  system  has  teeth  with  straight  sides,  the  two 
sides  of  a  tooth  making,  together,  an  angle  of  29°  (twice 


This  applies  to  gears  having  30  teeth  or  more.  For  gears 
having  less  than  30  teeth  special  rules  are  followed,  which  are 
explained  in  our  "  Practical  Treatise  on  Gearing." 


Fig.  2. 

In  epicycloidal,  or  double-curve  teeth,  the  formation  of  the 
curve  changes  at  the  pitch  circle.  The  outline  of  the  faces  of 
epicycloidal  teeth  may  be  traced  by  a  point  in  a  circle  rolling 
on  the  outside  of  pitch  circle  of  a  gear,  and  the  flanks  by  a  point 
in  a  circle  rolling  on  the  inside  of  the  pitch  circle.  The  faces 
of  one  gear  must  be  traced  by  the  same  circle  that  traces  the 
flanks  of  the  engaging  gear. 

In  our  practice  the  diameter  of  the  rolling  or  describing 
circle  is  equal  to  the  radius  of  a  i5-tooth  gear  of  the  pitch 
required  ;  this  is  the  base  of  the  system.  The  same  describing 
circle  being  used  for  all  gears  of  the  same  pitch. 


PROVIDENCE,    R.    I.  3 

The  teeth  of  the  rack  of  this  system  have  double  curves, 
which  may  be  traced  by  the  base  circle  rolling  alternately  on 
each  side  of  the  pitch  line. 

An  advantage  of  the  involute  over  the  epicycloidal  tooth  is, 
that  in  action  gears  having  involute  teeth  may  be  separated  a 
little  from  their  normal  positions  without  interfering  with  the 
angular  velocity,  which  is  not  possible  in  any  other  kind  of 
tooth. 

The  obliquity  of  action  is  sometimes  urged  as  an  objection 
to  involute  teeth,  but  a  full  consideration  of  the  subject  will 
show  that  the  importance  of  this  has  been  greatly  over-esti- 
mated. 

The  tooth  dimensions  for  both  the  involute  and  epicycloidal 
gears  may  be  calculated  from  the  formulas  in  Chapter  II. 


BROWN    &    SHARPE    MFG.    CO. 


n. 


SPUR    GEARING. 

(Figs.  *,  4.) 

Two  spur  gears  in  action  are  comparable  to  two  correspond- 
ing plain  rollers  whose  surfaces  are  in  contact,  these  surfaces 
representing  the  pitch  circles  of  the  gears. 

PITCH  OF  GEARS. 

For  convenience  of  expression  the  pitch  of  gears  may  be 
stated  as  follows  : 

Circular  pitch  is  the  distance  from  the  center  of  one  tooth  to 
the  center  of  the  next  tooth,  measured  on  the  pitch  line. 

Diametral  pitch  is  the  number  of  teeth  in  a  gear  per  inch  of 
pitch  diameter.  That  is,  a  gear  that  has,  say,  six  teeth  for  each 
inch  in  pitch  diameter  is  six  diametral  pitch,  or,  as  the  expres- 
sion is  universally  abbreviated,  it  is  "  six  pitch."  This  is  by 
far  the  most  convenient  way  of  expressing  the  relation  of 
diameter  to  number  of  teeth. 

Module  is  the  pitch  diameter  of  a  gear  divided  by  the 
number  of  teeth. 

Chordal  pitch  is  a  term  but  little  employed.  It  is  the  dis- 
tance from  center  to  center  of  two  adjacent  teeth  measured  in 
a  straight  line. 


Gear   Tooth    i    F>. 

Ctiordal  Thickness  of  Teetn  for  Gears  on 
a  Basis  of  i  Diametral  Pitch. 

S=Distance  from  pitch  line  to  top- of  teeth. 

S  Corrected=H+S. 

N= Number  of  teeth  in  gear. 

T=Chordal  thickness  of  Tooth.         T=D'  sin.  ft' 

H  =  Height  of  Arc.  H  =  R  ( i— cos.  ft') 

D'=  Pitch  Diameter. 

R=  Pitch  Radius. 

^'=90°  divided  by  the  number  of  teeth. 

NOTE— When  tin1  tooth  of  H  Rear  is  measured,  add  the  height  of  arc  to  iSi. 


Chordal   Thickness 


OF 


GEAR   TEETH. 


The  dimensions  of  Tooth  Parts  as  given  in  the  tables,  pages  6  to  9, 
are  correct  according  to  the  definition  of  Tooth  Parts  ;  but,  as  the  pitch 
line  of  gears  is  curved,  the  thickness  of  a  tooth  will  not  be  measured  on 
the  pitch  line  if  the  Caliper  is  set  to  the  figures  given  in  Tables  of  Tooth 
Parts. 


To  measure  the  tooth  accurately  the 
Caliper  must  be  set  to  the 

Chorclal 

See  Formula  on  reverse  page 


Gear 


Tooth    Calipei 


PROVIDENCE,    R.    I. 

FORMULAS. 

N  =  number  of  teeth. 

s  =  addendum   and  module. 

/  =  thickness  of  tooth  on  pitch  line. 

/=  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  tooth. 
D"  +  /  =  whole  depth  of  tocJi. 

d—  pitch  diameter. 
(f  =  outside  diameter. 
P'  =  circular  pitch. 
P^  =  chord  pitch. 

P  =  diametral  pitch, 
C  =  center  distance. 


P_N 


P'  =  - 


N        N  +  2 


2  2   P 

10 

D"  =  2  S 

p-I^lf 

* 

P'  =  dn where  sin  d 


d'  -  d  +  2  s 


BROWN    &    SHARPE   MFG.    CO. 


GEAR  WHEELS. 


TABLE   OF    TOOTH   PARTS CIRCULAR    PITCH   IN    FIRST    COLUMN. 


Circular 
Pitch. 

Threads  or 
Teeth  per  inch 
Linear  . 

Diametral 
Pitch. 

Thickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

Working  Depth 
of  Tooth. 

&   o5 

213 

121 

§<  £ 
« 

Whole  Depth 
of  Tooth. 

=L 

fiw 
Fj. 

PI 

•oS 

a  is 

T3  TJ 

*l 

P' 

i" 
f 

P 

t 

s 

D" 

•+/ 

D"+/ 

P'X.31 

P'X.335 

2 

i 

2 

1.5708 

1.0000 

.6366 

1.2732 

.7366 

1.3732 

.6200 

.6700 

If 

8 
IB" 

1.6755 

.9375 

.5968 

1.1937 

.6906 

1.2874 

.5813 

.6281 

H 

4 

T 

1.7952 

.8750 

.5570 

1.1141 

.6445 

1.2016 

.5425 

.5863 

H 

8 

laT 

1.9333 

.8125 

.5173 

1.0345 

.5985 

1.1158 

.5038 

.5444 

11 

2 
3 

2.0944 

.7500 

.4775 

.9549 

.5525 

1.0299 

.4650 

.5025 

Ifr 

16 
23 

2.1855 

.7187 

.4576 

.9151 

.5294 

.9870 

.4456 

.4816 

11 

JL 
11 

2.2848 

.6875 

4377 

.8754 

.5064 

.9441 

.4262 

.4606 

11 

3 
4 

2.3562 

.6666 

.4244 

.8488 

.4910 

.9154 

.4133 

.4466 

1* 

16 
21 

2.3936 

.6562 

4178 

.8356 

.4834 

.9012 

.4069 

.4397 

H 

4 
5 

2.5133 

.6250 

.3979 

.7958 

.4604 

.8583 

.3875 

.4188 

i* 

16 
19 

2.6456 

.5937 

.3780 

.7560 

.4374 

.8156 

.3681 

.3978 

il 

8 
9 

2.7925 

.5625 

3581 

.7162 

.4143 

.7724 

.3488 

.3769 

i* 

if 

2.9568 

.5312 

3382 

.6764 

.3913 

.7295 

.3294 

.3559 

i 

1 

3.1416 

.5000 

3183 

.6366 

.3683 

.6866 

.3100 

.3350 

15 
10 

1* 

3.3510 

.4687 

2984 

.5968 

.3453 

.6437 

.2906 

.3141 

7 
8 

1  1 

AT 

3.5904 

.4375 

2785 

.5570 

.3223 

.6007 

2713 

.2931 

£ 

ia 

3.8666 

.4062 

2586 

-.5173 

.2993 

.5579 

.2519 

.2722 

f 

11 

3.9270 

.4000 

2546 

.5092 

2946 

.5492 

2480 

.2680 

_L 

4 

n 

4.1888 

.3750 

2387 

.4775 

2762 

.5150 

2325 

.2513 

11 

16 

if 

4.5696 

.3437 

2189 

.4377 

.2532 

.4720 

2131 

.2303 

2 
3 

il 

4.7124 

.3333 

2122 

.4244 

2455 

.4577 

2066 

.2233 

5 

8 

il 

5.0265 

.3125 

1989 

.3979 

2301 

.4291 

1938 

.2094 

3 

'  5 

il 

5.2360 

.3000 

1910 

.3820 

2210 

.4120 

1860 

.2010 

JL. 

7 

if 

5.4978 

.2857 

1819 

.3638 

2105 

.3923 

1771 

.1914 

J_ 
16 

11 

5.5851 

.2812 

1790 

.3581 

2071 

.3862 

1744 

.1884 

PROVIDENCE,    R.    I. 


TABLE  OF  TOOTH  PARTS. — Continued. 


CIRCULAR    PITCH    IN    FIRST    COLUMN. 


Circular 
Pitch. 

Threads  or 
Teeth  per  inch 
Linear. 

Diametral 
Pitch. 

Thickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

l~ 

fil 

bo  o 
|H 

fo 

£ 

o3     $ 

«i*'\ 
<~  J2  ^ 

0  ^  A 

-d  r°   0 

J* 

£ 
«1 

3* 

*!* 

li« 

•1-1  O>  -M 

*r 

*l 

3^ 

§1 
£  2 

i 

P' 

i" 
"p7 

P 

t 

s  s 

D" 

•+/ 

D'4-/. 

PX.31 

PX.335 

1 

2 

6.2832 

.2500 

.1592 

.3183 

.1842 

.3433 

.1550 

.1675 

k 
e 

2| 

7.0685 

.2222 

.1415 

.2830 

.1637 

.3052 

.1378 

.1489 

7 
1C 

2f 

7.1808 

.2187 

.1393 

.2785 

.1611 

.3003 

.1356 

.1466 

A 

7 

21 

7.3304 

.2143 

.1364 

.2728 

.1578 

.2942 

.1328 

.1436 

2 

T 

21 

7.8540 

.2000 

.1273 

.2546 

.1473 

.2746 

.1240 

.1340 

3 
8 

2f 

8.3776 

.1875 

.1194 

.2387 

.1381 

.2575 

.1163 

.1256 

4 

11 

2f 

8.6394 

.1818 

.1158 

.2316 

.1340 

.2498 

.1127 

.1218 

1 
T 

3 

9.4248 

.1666 

.1061 

.2122 

.1228 

.2289 

.1033 

.1117 

5 
16 

31 

10.0531 

.1562 

.0995 

.1989 

.1151 

.2146 

.0969 

.1047 

To 

31 

10.4719 

.1500 

.0955 

.1910 

.1105 

.2060 

.0930 

.1005 

2 

7 

31 

10.9956 

.1429 

.0909 

.1819 

.1052 

.1962 

.0886 

.0957 

1 
i 

4 

12.5664 

.1250 

.0796 

.1591 

.0921 

.1716 

.0775 

.0838 

T 

41 

14.1372 

.1111 

.0707 

.1415 

.0818 

.1526 

.0689 

.0744 

T 

5 

15.7080 

.1000 

.0637 

.1273 

.0737 

.1373 

.0620 

.0670 

3 
10 

5i 

16.7552 

.0937 

.0597 

.1194 

.0690 

.1287 

.0581 

.0628 

2 
11 

51 

17.2788 

.0909 

.0579 

.1158 

.0670 

.1249 

.0564 

.0609 

1 

T 

6 

18.8496 

.0833 

.0531 

.1061 

.0614 

.1144 

.0517 

.0558 

2 
13 

61 

20.4203 

.0769 

.0489 

.0978 

.0566 

.1055 

.0477 

.0515 

1 

7 

7 

21.9911 

.0714 

.0455 

.0910 

.0526 

.0981 

.0443 

.0479 

2 

15 

71 

23.5619 

.0666 

.0425 

.0850 

.0492 

.0917 

.0414 

.0446 

1 

T 

8 

25.1327 

.0625 

.0398 

.0796 

.0460 

.0858 

.0388 

.0419 

1 

9 

9 

28.2743 

.0555 

.0354 

.0707 

.0409 

.0763 

.0344 

.0372 

1 

10 

10 

31.4159 

.0500 

.0318 

.0637 

.0368 

.0687 

.0310 

.0335 

1 
10 

16 

50.2655 

.0312 

.0199 

.0398 

.0230 

.0429 

.0194 

.0209 

1 

20 

20 

62.8318 

.0250 

.0159 

.0318 

.0184 

.0343 

.0155 

.0167 

BROWN    &    SHARPE    MFG.    CO. 


GEAR  WHEELS. 

TABLE  OF  TOOTH  PARTS DIAMETRAL    PITCH   IN    FIRST    COLUMN. 


Diametral 
Pitch. 

Circular 
Pitch. 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

Working  Depth 
of  Tooth. 

09 

fl 

Q 

I* 

P 

P' 

t 

s 

D" 

«+/. 

D"+/. 

i 

6.2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

I 

4.1888 

2.0944 

1.3333 

2.6666 

1.5428 

2.8761 

1 

3.1416 

1.5708. 

1.0000 

2.0000 

1.1571 

2.1571 

1J 

2.5133 

1.2566 

.8000 

1.6000 

.9257 

1.7257 

!i 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

If 

1.7952 

.8976 

.5714 

1.1429 

.6612 

1.2326 

2 

1.5708 

.7854 

.5000 

1.0000 

.5785 

1.0785 

2J 

1.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

2i 

1.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2f 

1.1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

1.0472 

.5236 

.3333 

.6666 

.3857 

.7190 

8J 

.8976 

.4488 

.2857 

.5714 

.3306 

.6163 

4 

.7854 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

6 

.5236 

.2618 

.1666 

.3333 

.1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927 

.1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

.2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

.1659 

14 

.2244 

.1122 

.0714 

.1429 

.0826 

.1541 

PROVIDENCE,    R.    I. 


TABLE  OF  TOOTH  PAKTS—  Continued. 


DIAMETRAL    PITCH    IN    FIRST    COLUMN. 


M  Diametral 
Pitch. 

Circular 
Pitch. 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

Working  Depth 
of  Tooth. 

rv    <U 

"** 

0^ 
&>  — 

^^  £ 

&  £ 

Q 

tl 
SI 

0«^ 

r3  ° 
£ 

P'. 

t. 

8. 

D". 

«+/. 

D"  +  / 

15 

.2094 

.1047 

.0866 

.1333 

.0771 

.1438 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

17 

.1848 

.0924 

.0588 

.1176 

.0681 

.1269 

18 

.1745 

.0873 

.0555 

.1111 

.C643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

.1208 

.0604 

.0385 

.0769 

.0445  • 

.0829 

28 

.1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

32 

.0982 

.0491 

.0312 

.0625 

.0362 

.0674 

34 

.0924 

.0462 

.0294 

.0588 

.0340 

.0634 

36 

.0873 

.0436 

.0278 

.0555 

.0321 

.0599 

38 

.0827 

.0413 

.0263 

.0526 

.0304 

.0568 

40 

.0785 

.0393 

.0250 

.0500 

.0289 

.0539 

42 

.0748 

.0374 

.0238 

.0476 

.0275 

.0514 

44 

.0714 

.0357 

.0227 

.0455 

.0263 

.0490 

46 

.0683 

.0341 

.0217 

.0435 

.0252 

.0469 

48 

.0654 

.0327 

.0208 

.0417 

.0241 

.0449 

50 

.0628 

.0314 

.0200 

.0400 

.0231 

.0431 

56 

.0561 

.0280 

.0178 

.0357 

.0207 

.0385 

60 

.0524 

.0262 

.0166 

.0333 

.0193 

.0360 

10 


BROWN    &    SHARPE    MFG.    CO. 


Comparative  Sizes  of  Gear  Teeth. 
Involute. 


8  P 


Fig.  4. 


PROVIDENCE,    R.    I. 


IT 


CHAPTER     III. 

BEVEL  GEARS.— AXES  AT  RIGHT  ANGLES. 

(Fig.  5.) 


12  BROWN    &    SHARPE   MFG.    CO. 


FORMULAS. 

N-=   [Number  of  teeth  j 

P  =  diametral  pitch. 
P'  =  circular  pitch. 

aa  =   }  center  angle  =  angle  of  edge  j  gear. 
ab  —   \  or  pitch  angle  (  pinion. 

ft  =  angle  of  top. 
fi'  —  angle  of  bottom. 

g  =  [angle  of  face 


A  =  apex  distance  from  pitch  circle. 
A'  =  apex  distance  from  large  bottom  of  tooth. 

d  =  pitch  diameter. 
d'  =  outside  diameter. 

s  =  addendum   and  module. 

/  =  thickness  of  tooth  at  pitch  line. 
/  =  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  tooth. 
D"  +  /=  whole  depth  of  tooth. 
2  a  =  diameter  increment. 

b  —  distance  from  top  of  tooth  to  plane  of  pitch  circle, 
F  =  width  of  face. 


PROVIDENCE,    R.    I.  13 


tan  «.=-*-•-;       tan  «.  = 


;or      tan  /?  =     - 

N  A 


tan  /S'  =  sng2  +  T      ==  2.314  sn  «   .     tan  ^  = 


a  -  90°  -  K  +  ft)  \gb  =  90°  - 
=  a—  ft'         (See  Note,  page  69.) 


A= 


2  P  sin  a 

N 


A'=      ^  A'  = 

cos  p'  2  P  sin  a  cos 

A',=  -  - *—  -,  cos  ft 
sin  («•  +  /?) 

P,         N 


2  A  sin  a 


P  7T 

2  a  =  2  s  cos  <*  (Seepage  20.) 

a  for  gear      =  b  for  pinion 


tf  for  pinion  =  *  for  gear 

p  _      ?T_ 
- 


J=  JL  =        =  .3183?'     j  =  A  tan/? 

P  71 

s  +  /  =  .  3685  P'  s  +  /  =  A  tan  ft' 


N~  N~  ~A~ 


F  =  1  +  _  or  =  2  P'  to  3  P' 
P       7 

NOTE. —Formulas  containing  notations  without  the  designating  letters  a  and  b 
apply  equally  to  either  gear  or  pinion.  If  wanted  for  one  or  the  other,  the  respective 
letters  are  simply  attached. 


BROWN    &    SHARPE   MFG.    CO. 


BEVEL  GEARS  WITH  AXES  AT  ANY  ANGLE. 


Pinion 


Fly.  6. 


PROVIDENCE,    R.    I, 


FORMULAS. 

C  =  angle  formed  by  axes  of  gears. 
N;  =   }  number  of   teeth  )  ^on. 

P  =  diametral  pitch. 
P'  =  circular  pitch. 

'   "l=\  an£le  of  edSe  =  Pitch  anSle  |  pfnior  . 
ft  =  angle  of  top. 
ft  —  angle  of  bottom. 

£  =   [angle  of  face  |  S^on. 
£  =   [cutting  angle  \^on 

A  =  apex  distance  from  pitch  circle. 
A'  =  apex  distance  from  large  bottom  of  tooth. 

d=  pitch  diameter. 
d'  =  outside  diameter. 
2  a  =  diameter  increment. 

b  =  distance  from  top  of  tooth  to  plane  of  pitch  circle. 

NOTE.— The  formulas  for  tooth  parts  as  given  on  page  5  apply  equally  to  these 
cases. 


1 6  BROWN    &    SHARPE    MFG.    CO. 


tan  aa  =  —  -^  -  ;  or  cot  aa  =  —  —  -—  4-  cot  C 
^  +  cos  C  Na  sm  C 

Na 

sin  C  N« 

tan  txb  =  —  —  -  ;  or  cot  ab  =  —  —  ^—  +  cot  C 

N,,,  N6  sm  C 


NOTE.  —  The  above  formulas  are  correct  only  for  values  of  C  less  than  90 
If  C  is  greater  than  90°,  consult  page  18. 


a        2  sm  a  s 

tan  p  -     —  -  —  ;     or  tan  p  =  —  ; 


tan  ft'  =  **«+=  2.314  sn  nr  .  tan  ft,  =  5+    . 

£•„  —  9o°  —  (^tt  +  /?)  for  Cases  I  and  II. 

ga  =  ft,  for  Case  III. 

ga  —  9o°  —(«-„  —  /?)  for  Case  IV. 

5-6  =  90°—  («-*,+/?) 

h  —  a  —  ^  (See  page  69.  ) 


N 


f  for  Cases  I  and  II, 

a    =•  a  -\-  z  a  \  .    .         .  TTT  TTT 

I  and  pinions  in  Cases  III  and  IV. 

d'  =  d,  for  gear  in  Case  III. 

d'  =  d  —  2  a,  for  gear  in  Case  IV. 

2  a  -=2s  cos  a 

b  =  s  sin  # 

NOTE. — Formulas  containing  notations  without  the  designating  letters  a  and  t> 
apply  equally  to  either  gear  or  pinion.  If  wanted  for  one  or  the  other,  the 
respective  letters  are  simply  attached. 


PROVIDENCE,    R.    I. 


l8  BROWN    &    SHARPE    MFG.    CO. 


The  formulas  given  for  aa  and  ab  (when  C,  Na  and  N6  are 
known)  undergo  some  modifications  for  values  of  C  greater 
than  90°. 

For  bevel  gears  at  any  angle  but  90°  we  may  distinguish 
four  cases  ;  C,  N0,  N6  being  given. 

/.  Case.     See  pages  14  and  16. 

II.  Case.     C  is  greater  than  90°. 

sin  (180  —  C)  sin  (180  —  C 

tan  ofa=  ——  —L-  ;     tan  ab  =  __ 

_ 6-cos(i8o-C)  —a-cos(iSo-C} 

Na  N6 

///.  Case.     (xa  =  90°  ;  ab  =  C  —  90° 
IV.  Case. 

sinE  sin  E 


1ST  NT 

cos  E  -  1N-&  _a  -  cos  E 

Na  N6 

For  an  example  to  apply  to  Case  III.,  the  following  condi- 
tion must  be  fulfilled  : 

Na  sin  (C  -  90°)  =  Nb 

To  distinguish  whether  a  given  example  belongs  to  Case  II. 
or  case  IV.,  we  are  guided  by  the  following  condition  : 

T       ^j     •     /p.  0\  j  smaller  than  N6,  we  have  Case  II. 

iMasm  ({,  —  90  )  -j  ]„„„„„  +v,an  XT     «r^  v,o^^  case  IV. 


PROVIDENCE,  R.  I.  19 


UNDERCUT  IN  BEVEL  GEARS. 

By  undercut  in  gears  is  understood  a  special  formation  of 
the  tooth,  which  may  be  explained  by  saying  that  the  elements 
of  the  tooth  below  the  pitch  line  are  nearer  the  center  line  of 
the  tooth  than  those  on  the  pitch  line.  Such  a  tooth  outline  is 
to  be  found  only  in  gears  with  few  teeth.  In  a  pair  of  bevel 
gears  where  the  pinion  is  low-numbered  and  the  ratio  high,  we 
are  apt  to  have  undercut.  For  a  pair  of  running  gears  this 
condition  presents  no  objection.  Should,  however,  these  gears 
be  intended  as  patterns  to  cast  from,  they  would  be  found  use- 
less, from  the  fact  that  they  would  not  draw  out  of  the  sand. 
We  have  stated  on  page  2  (see  Fig.  i)  that  the  base  of  our 
involute  system  is  the  14^°  pressure  angle.  If  a  pair  of  bevel 
gears  with  teeth  constructed  on  this  basis  have  undercut,  we 
can  nearly  eliminate  the  undercut — and  for  the  practical  work- 
ing this  is  quite  sufficient— by  taking  as  a  basis  for  the  con- 
struction of  the  tooth  outline  a  pressure  angle  of  20°. 

The  question  now  is  :  When  do  we,  and  when  do  we  not 
have  undercut  ?  Let  there  be  : 

N  =  number  of  teeth  in  gear. 
n  =  number  of  teeth  in  pinion. 


n*  =  4 


where  we  have  undercut  for/  less  than  30. 

This  formula  is  strictly  correct  for  epicycloidal  gears  only. 
It  is,  however,  used  as  a  safe  and  efficient  approximation  for 
the  involute  system. 


20 


BROWN    &    SHARPE    MFG.    CO. 


DIAMETER    INCREMENT. 

2  a. 

RULE. — The  ratio  being  given  or  determined,  to  find  the  outside  diameter 
divide  figures  given  in  table  for  large  and  small  gear  by  pitch  (P;  and  add 
quotient  to  pitch  diameter. 


RATIO. 

GEARS. 

RATIO. 

GEARS. 

RATIO. 

GEARS. 

Large 

Small 

Large 

Small 

Large 

SmalJ 

1  00 

1:1 

.41 

1.41 

1   65 

1.05 

1.70 

4.40 

.45 

1.94 

1.05 

.37 

1.42 

1.67 

5:3 

1.03 

1.72 

4  50 

9:2 

.44 

1.95 

1.07 

.36 

.43 

1.70 

1.01 

1.73 

4.60 

.42 

1  95 

1.10 

.35 

.44 

1.75 

7:4 

.99 

1.74 

4.80 

.41 

1.96 

1.11 

10:9 

.34 

.46 

1.80 

9:5 

.97 

1.75 

5  00     5:1 

.39 

1.96 

1.12 

.33 

46 

1.85 

.95 

1.76 

5.20 

.38  |  1.96 

1.13 

9:8 

.33 

.47 

1  90 

.93 

1  77 

5.40 

.37     1.96 

1.14 

8:7 

.32 

.49 

1  95 

.91 

1.78 

5.60 

.36 

1.97 

1.15 

1.31 

.50 

2  00 

2:1 

.89 

.79 

5.80 

.34 

1.97 

1.16 

1.30 

.51 

2.10 

.87 

.80 

6.00 

6:1 

.33 

1.97 

.17 

7:6 

1.30 

.52 

2.20 

.84 

.81 

6.20 

.32 

1  97 

.18 

1.29 

1.53 

2  25 

9:4 

.82 

.82 

6.40 

.31 

1.97 

.19 

1.28 

1  53 

2.30 

.80 

.83 

6.60 

.30 

1  97 

.20 

6:5 

1.28 

1  54 

2.33 

7:3 

.78 

.84 

6.80 

.29 

1  98 

.23 

1.27 

1.55 

2.40 

.76 

.85 

7  00 

7:1 

.28     1.98 

.25 

5:4 

1.25 

1.56 

2.50 

5:2 

.75 

86 

7.20 

.27 

1.98 

.27 

1.25 

1.57 

2.60 

.73 

.86 

7.40 

.27 

1  98 

.29 

9:7 

1.24 

1.58 

2.67 

8:3 

.71 

.87 

7.60 

.26     1  98 

1.30 

1.22 

1.59 

2.70 

.69 

.87 

7  80 

.26     1.98 

1.33 

4:3 

1.20 

1.60 

2.80 

.67 

.88 

8  00 

8:1 

.25     1.98 

1.35 

1.18 

1.61 

2.90 

.65 

1.89 

8.20 

.24      1.98 

1  37 

1.17 

1.61 

3.00 

3:1 

.63 

1.91 

8  40 

.24     1.98 

1.40 

7:5 

1.16 

1.62 

3.20 

.60 

1.92 

8.60 

.23     1.98 

1.43 

10:7 

1.15 

1.63 

3.33 

.58 

1  92 

8.80 

.23     1.98 

1.45 

1.13 

1.65 

3.40 

.56 

1  92 

9.00 

9:1 

.22     1.99 

1.50 

3:2 

1.11 

1.66 

3.50 

7:2 

.54 

1.93 

9.20 

.22     1.99 

1.53 

1.10 

1.67 

3.60 

.52 

1  93 

9.40 

.21      1.99 

1  55 

1.09 

1  67 

3  80 

.50 

1.94 

9.60 

.21     2.00 

1.58 

1.08 

1.68 

4.00 

4:1 

.49 

1.94 

9.80 

.20     2.00 

1.60 

8:5 

1.07 

1.68 

4.20 

.47 

1.94 

10.00    10:1 

.20  i  2  00 

1 

NOTE. — To  be  used  only  for  bevel  gears  with  axes  at  right  angle. 


PROVIDENCE,    R.    I.  21 


TABLES  FOR  ANGLES  OF  EDGE  AND  ANGLES 

OF  FACE. 

The  following  four  tables  have  been  computed  for  the 
convenience  in  calculating  datas  for  bevel  gears  with  axes  at 
right  angle.  They  do  not  hold  QQQ>&  for  bevel  gears  with  axes 
at  any  other  angle. 

To  use  the  tables  the  number  of  teeth  in  gear  and  pinion 
must  be  known. 

Having  located  the  number  of  teeth  in  the  gear  on  the 
horizontal  line  of  figures  at  the  top  of  the  table,  and  the  num- 
ber of  teeth  in  the  pinion  on  the  vertical  line  of  figures  on  the 
left-hand  side,  we  follow  the  two  columns  to  the  square  formed 
by  their  intersections. 

The  two  angles  found  in  the  same  square  are  the  respective 
angles  for  gear  and  pinion.  The  tables  are  so  arranged  that 
the  angle  belonging  to  the  gear  is  always  placed  above  the 
angle  for  the  pinion. 


22 


BROWN    &    SHARPE    MFG.    CO. 


TABLE  i 

ANGLE  OF  EDGE. 
GEAR. 


41 

40|39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

12 

73°4i 

73*18  paV 

72*38 

72"e 

7  13* 

71  's' 

703+ 

70'  i' 

69°26 

63°50' 
2lV 

68V 

2  1  *48 

6731 
22  M 

6648 

23*12 

66V 
23*s» 

6*19 

6V  17V 

1732 

17  58 

1886 

18*55 

1926 

1359 

^034 

13 

78*25 
17*35 

71  59 

I8*»' 

71  M 

1  8*26 

71*7' 
18*53' 

/I)  39 

I9*ef 

70°  9 

19*51' 

69*37' 

20*23' 

GS's 

ao'ss 

6830 

2  1  *3li 

67°53 
22*7' 

67'|6 

22*45 

66*34 

23*86 

65"sr 

24*9 

65*6 

24*54 

64*17' 

25*43 

14 

71    9 

I8*5l' 

7043 
13*17 

70  is 

19*45 

6946 

eoV 

69  16 

20*44 

68°4S 

21*5 

68*12' 

21*48 

6737 
22*Z3 

6/  o 
23°o 

6623 
63*37 

6S°«' 
24*m' 

64*59 
25*  T 

64°K 

25*46 

63*26 

26V 

62'36 

27*24 

15 

69*54 

2oV 

69*e668°s8 
20*3412  iV 

68*88 

21  W 

67  56 

22*4 

67*23 

22*37' 

66*48 
23  M 

66*12 

23*48 

65  »3 

24*27 

64*53 

esV 

64  V 
25*so' 

63*2t 

26*34' 

62*39 

27°2i 

C.I49 
28*n' 

60*S7 
29°J1 

16 

68V 
21  V 

68V 

8148 

674! 

22*16 

67*0 
22*5d 

6637 

23*E3 

662' 

23*58 

65°26 
24*34 

6448 
2S°I2 

64"  8 

25*58 

63*e& 

26M 

62V 
27*« 

6l*Sfc' 
28*4' 

61*7 
28« 

60ai5 

29*45 

59*2i' 

30*33' 

17 

6729 

66*« 
23Y 

66  27 

65V 

65'» 

6*43 

64"  6 

6326 

6245 

62%' 

61  "is 

eo'tt 

59"37 

584* 

5748' 

2333 

U46 

244i 

25'i7 

25  54 

2634 

27  is 

2/59 

28*45 

29  3z 

30  n 

31  16 

32  iz 

18 

66"ie 

2348 

654* 
24°w 

65V 
24%6 

64*S9 
25V 

fa4J4 
25*ES 

fea1?*' 
2b*M 

6247 

27°  n 

b26 

87*54 

6l't3 

28*3?' 

60  36 
29*22 

5951 
30*9 

59*2' 

30°58 

58'w 
3i*w 

57l|6 

32*44 

56*19 
33V 

19 

65*8 
24*si 

64*1* 

64"e 

6326 

68°49 

62'  10 

61*30 

6048 

60'4 

59Jm 

58"3d 

5739 

56« 

55si' 

54*52 

2S«4 

B5*S8 

26»» 

C7  11 

27  so 

28*30 

29  ti 

2956 

30V 

31*30 

32  ei 

A3  (4 

34  9 

35'8' 

PINION. 

20 

164V 
26*o' 

63'2«  62*51 
26*3*'  27V 

62"i4 

6I"S7 

60US7 

60UiS 

59J3£' 

5847' 

58V 

57UK> 

56°  19 

55°24 

54"28 

53*28 

3028 

JU  o 

3250 

33  41 

i436 

3b32 

3637 

21 

68*53 

27*7' 

62*8  6I°42 
27°4d  28*18 

61*4' 
28°56 

60"es 

29*35 

sa^' 

30*is 

59*  £' 

30*58' 

58"« 

31*42 

57L32 

32*28 

56  '43 
33*17 

55-53 
34*7' 

55Jo 
35*o' 

54-5' 

35*55 

53*7 

36*53' 

52*8 
37sz' 

22 

61*47 

61*  «' 

60*34 

53S6 

59  IS 

58°34 

5751 

576' 

56*13 

55yz9 

54*38 

53*45 

52"43 

51*50 

50*49' 

2f£i3 

2849 

29  ?6 

304 

30  45 

3)26 

329 

32  S4 

3341 

3431 

3522 

36i5 

37H 

38  10 

39)1 

23 

60V 

60V 

59*28 

5849 

58*8 

57*25 

scV 

55*55 

5SV 

54°  e 

53*26 

52*3.' 

51*35 

5036 

4^34 

eg5* 

29°S4J3032 

31  ti 

31  52 

32  35 

33  13 

34s 

3453 

3542 

363* 

37*89 

3&8S 

3924 

4026 

24 

S9J9 
30V 

59Y 
30  SB 

5823 

57  '44 

57le 

5619 

55U33 

5447 

53*58 

53*7 

52*15 

5I°20 

SOa 

43  24 

48« 

31  37 

32  it 

32  sa 

334i 

3427 

3bi3 

36*2 

3653 

3745 

38*40' 

3937 

4036 

41  3* 

25 

se» 

3lV 

seV 

32V 

5726 
%4o 

56*40 
33*eo' 

55*57 

34*a 

ssV 

34*47 

54*88 
35*32' 

53« 
36*20 

52*st 
37*3* 

52V 
38V 

51*7' 
38*53' 

50°i2 
39°48' 

49V 

40*46 

48"  14 
4146 

47*l£ 

42*4« 

26 

157*37 

5658 

56"  19 

5537 

54*54 

54°  16 

53M 

52"3fr 

51*46 

50JS4 

soV 

49*  & 

48U7 

47U7 

46*5 

3223 

33  e 

3341 

3423 

3b6 

35so 

36  3* 

3/24 

38)4 

396 

39  59 

4055 

4153 

4253 

43  K 

27 

56"JS 

5559 

ss"«j 

54U36 

53"S3 

53U7 

52*21 

51  ^ 

SO'in 

49U5I 

48U57 

4«V 

473' 
42*57 

46*2 
43*58 

4-5* 

33  a 

34i 

344? 

3!>24 

367 

3bs3 

3/39 

38e? 

3317 

409 

41   3 

42o 

28 

55*w 
34*» 

55> 

35  o' 

54*19 

3SV 

53"37 

52"w 

52J8 

5l*zo 

5032 

49°4l 

48"48 

47U5S 

46^58 

46*0 

45* 

3623 

3/7 

3/sz 

3S40 

&»<* 

40  19 

41  12 

42  S 

43a 

44"  o 

29 

54V 

54*3 

53a 

S2139 

5l°55 

51*9 

soV 

43J32 

40*28 

48°4i 
4I*» 

4749 
42*  »' 

46V 
43*6' 

45^8 
44*z' 

45* 

35  16 

35*57  36*38 

37  ei 

38  s 

38  s> 

3339 

30 

153*4* 
36*,2 

S3*7|S2*t6 
36°S3]37V 

5142' 

50°S8 

SO'e 

4924 

483S 

47« 

46°5I 

45"56 

45° 

38)8 

39  « 

ddw 

40  M 

4(25 

4217 

439 

44*4 

31 

5254 
37V 

52°i3]5lV 

50°48 

50Y 

49",6 

48°28 

47» 

4647 

45JS4 

45* 

3747  3*W 

39  w 

39  58 

4044 

4-1  '32 

4221 

43)3 

446 

32 

52'z 

5  1'wSO'w 

49*54 

49"  9 

48°H 

47°34 

46*44 

4SJ5J 

45* 

37  ss 

384039*88 

40  si 

41*38 

42  26 

43  16 

447 

«X 

51*10 

50*E9l49°46 

49°2 

48  16 

47*« 

4641 

45si 
44*9 

45* 

«s\ 

38so 

39°3i  40*4 

40  z» 

41  44 

42zi 

43°i» 

34 

50*20 

49~»48°SS 

48*»' 

47*25 

<K>"3« 

4S"so 

45* 

3940 

40°»'4lV 

4I°49 

4V>35 

43622 

44)0' 

35 

49*3i 

48%8|4e*s 

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46"35 

45<4S 

45" 

4023 

4l°ie 

41  55 

4239 

4345 

44  12 

36 

48*43 
41*17 

48"  0 
42°  o 

47U(7 

4243 

46*33 
43*27 

4547 

44*« 

45° 

37 

47» 
42*4 

47*14 

42*44 

46°3« 
43°3Q 

454* 
44V 

45° 

38 

47'io 

42*55 

46'w 
43-3S 

45"*s 
44-*is 

45* 

39 

46*26 
4334 

4543 
44*7 

45* 

40 

|45*4z 
44*,8 

45° 

4 

45' 

PROVIDENCE,    R.    I. 


TABLE   i. —(Continued^ 


ANGLE  OF  EDGE. 


GEAR. 


2625242322 


21 


20l9 


18 


17 


16 


5 


I4 


I3 


I2 


12 


si 


64*22 


6326 


6227 
27*33 


60.s 


2837 


2945  30  sal  32  16 


537 

36*53 


51  80492447  17 
38*40 


40°3C 


45 


\f* 

•« 


63  ee  6231 


56*561 5  5  37 

33 


34  23  35  so 


s 

4055 


47  7 
42*53 


45 


594S 
30*5 


5840 

31*20 


7  x  56  19  55  o 


32  28  33  41 


5337 

35"ol36*z3' 


52  8 

37°52 


43  e' 


45 


59*2 
30*58 


50V 


32o 


3532 


38  18  3848 


45° 


53  SB  58  42' 


51  °EO  43*5*  48°Z2 


43  16 


45* 


I75649 

I/   33  ii- 


5547 


5441 


5332 

36*28 


51*0* 


3742 


38  o' 


49  3*  48  ii 
40"zj  41*43 


18 


*I5 

35*4s 


SVv 


51*57 


4036 


4633 

42"»|4327 


53SI 
36*9 


5I°38 
38°E2 


SO°X  49n 


4043  4?  8 


20  SO  R 


45° 


2  I 

••  1 


38  se 


4958 
40*  z' 


4848' 


473*1 46*i 
43°40 


41*21 


42*3i 


43V 


4-5 


48  30  47  23 


47"17 


46'io 
43*so' 


45° 


45* 


BROWN    &    SHARPE    MFG.    CO. 


TABLE    2. 


ANGLE  OF  EDGE. 
GEAR. 


7271 


686766656463 


61 


60595857 


12 


8033 
9*«? 


80* 
9*36 


79*» 
I0*| 


79  6t  73 


79  3Z  79  23  79  13 


0*37    10*47 


78*52  70*1 


78  3d  78  19  78  7 


II  53 
77V 
I2*5i' 


13 


794* 
10*14 


7937 
.0*23 


7929  79zo 
10*31    10*40 


79*11 
10*49 


79  I 


1851 


784 


IOW 


78 31  78*20 
il*JMll*4< 


77°5«  77°46  77  • 


12  2 


14 


79o 
11*0 


78*si 

11*9 


7641  76°32 


78*22 
It*  38 


77*51 


II  59 


77-40  77*28 
12*20  12*32 


76* 

13*2/1 13*3* 


76*26  76  ti. 


348 

rs^i 

]4°45 

74°  i< 

15*4 


78% 


77V 

12*6 


77  44  77  3* 


7723 

12*37 


77*12 

12*48 


76*48  76  »e 


7S°58  7S*44]7S'3i  75*15 
14*  *  1430 


16  £ 


77*»  77*7 


76S7 


'2  S3  13  3 


1315 


763* 

13*26 


76*22 
13*38 


76  10 
I3°SO 


75*58 


75% 


74°49  74*35  74°  19 


I5"|||1S°2S 
73*56|  73*40 


17 


76*43 

13V 


76*» 
I3*» 


76*2i']76*id 
I3*39'|  13*50 


7S*» 

14"  z 


7S°4S  75  33 


758 


74*5* 


74°..' 


18 


7558 
14°  Z 


7546 
14*14 


75  » 
I4*2i'|  I4~37 


75  M  75*io 


74*56 

15*2 


7445 

15*15 


74*31 


74°  17  74*3 

15431  15*58 


73*ie 


16*58 


72*29 

12 

7  I 

1826 
70°40 

19*20 


19 


75*13 
»4*47 


75  r 

14*59 


74*49  74  3*  74*23 


15  II !  15  24 


74*  to 
15*50 


73*56 
16*4 


73*42 

16*18 


73  as)  73*t: 
16*32  16*47 


72*si 


7242 


17  16 


7220 
17*3* 


72*9   71*52 
I7*5l1  18*8 


|74*t»| 
15*3 


7416 

IS  44 


74*3 
I5*S7|  16"  10 


73°sd  73*37 
16*23 


73*9 
I6°si 


72*5* 
17*6 


72^72*23 
17*21    17*37 


72*7 


7I°5! 


1753 


71*16  7059 
18*4*  I9f 


21 


73  .8  734' 

I6°56 


72*sp 
17*  K) 


72°«  72  21 


17  J* 


17  is 


726 
17*5* 


71  I 


71°  o 


18  10  1826 


18  43 


19  o 


7O43 
19*  ,7 


70°z*'|70  6 


|73*T 
16-59 


72*47 

17*3 


72  19 
17*41 


72*4 
17°5« 


71*49 


71*34 


1826 


TIM 

18*43 


71*  i  70*45 


70*28  70*10 


6952  69"  33 i  69  13 


18  yj  \9  is 


1934 


1950 


68*54 

20'27|2047|2I*6 
68*4e68*t2|68*2 


72°a 

I7*$7 


I8n 


71*9 

18'si 


70V 


70  JO 
I9*» 


70*14^  69*57 
I9°46l20*3 


63  39  69°2C 


202! 


2040 


. 

18  tt 


71*19 
I8*4l' 


71*5    70*49 


70e34  70°  17 
I9°43 


70* 

I9°59 


89V 
20*16 


20*3*1 20°s 


68  50  68  3 
2l°id  2l*N 


68  iz 


67  52  67  3i 


21  48  22  8  \ZZZ9  22  50 


70*51' 
19*9 


70*36  70  ti'  70*5' 
I9*W  19*55 


69  49  69  J2  69°>5i  68V  68° 


20(1 


202B204S2I  3 


40  68  zi 
2l'*20 1  21*39 


33' 


67  43  67*23  67  Z  66* 


21  57  22  17 


22  37  2258  23  19  2341 


709 
19V 


693769V 


69*4-' 


207 


20*0  20*39  20* 


63  46  68*30  68  iz 


21*30 


2146 


22*6  22*» 


67  34  67  is  6655  663*'  66  13  6Ss 


2245  23  5 


2326  2347  249 


6529 

24*31 


27 


V  69*10 


66 54|68e38  6620  68*3 


67  4i  67°t6  67*8  66°4i 


20*50  21*6  21*22' 


21  57  22  IS 


22*34  22*52 


66*28  66*7 


65*46  65*25  65  2 


23*53  24*14  24*35 1 24*58  25*11 


68  45  68*29 

21*15  21*31 


68*11  67*55' 
21*4822*5' 


67V 
22°Z3  22 


67*19  67"  | 
22*59 


MM 
23*18  23 


fc&  Zi  64  53 
24°» 


63*sb 
25°2i|25*46  26*io 


6,747 


67* ti  66*s*  66*36  66*17  6S°S7  65*37  65*16  64  55  6434  64*iz  63*SO  63T26  Sfz 


22*48  23*6 


z    2343 


243 


25*5 


2S  26  25  4« 


26°io  26*3*  26°58 


6fe°4»  66*30  fetfit  65*51  65*33  6SV  64*si  64°32  64*  to  63*45  63°»  es'a' 


22  S7 


22V 


23*12  23*SO  J»°4«  24* 


2446  85  7 


25*28  25*50  26  I 


2634 


26*57  27*21 


2744 


31 


6«*4Z  66»' 


23t8 


23*35 


666 
23*1* 


65466596510 
24* li  24*31 


64  SO  6430  64  9 1 63  48  63  £«  63  3 


24*6i  25*10  2S°3(i  25*51  26* 


62**b 


16*3*  Zb'si  27*20' 


62  18  6I* 


6544  6S*»  65*7' 


23°5«i  24*16 


24*34  24*53  25V 


64*46  64*2*  ftt 


25322SS2 


6V*7  63 

26*13  26*34|26*56 


27°I8  27*41 


62*19  6I*M  61* 


28V 


617 
28*53 


60*4 
29°,9 
5»S6 


6!>Z}  65V 


24*37 


62^43  6Z2i  61  st  6l*3&|6ru 
24*56  25*isl2S*»4l 25*53  26*13  260S4|26°S5  27*7  27°» 


64*26  64*7 


6347  63  26  63  S 


6047|602i 

3 


28492913  2939304- 


34 


6443  fe4°25 
25*.7 


63*46' 
2555  26*>4 


265527 


6245 


62*13 


62*,'6. 
27  59 


60  52  60  88  60  3 


Z»*w  2845  29*8 


59V 
57  30*23' 


59*u 

30*49' 
58*27 
31*33 


63*4563' 


26*3426*54 


6246  6?.  25  62  4.' 


6I4Z  M 


27*14  27* JS  27*56  28*  18  2841  29*3 


t057  60T33  60  9 


59  *5  59  i9  58  si 


2927 


2951 


30  IS  3041  31  7 


t*A*1i*rd 

27*3  27  33 


62*6 


6145  6f 


*»ri 


27°S*  28*<5  28°37  28*59 


i|60  is  59  si 
30*9 


30*33 


«-  S837. 

30°58  31*23  31*50  32* ri 


4«  6228  62  8  61*48 


6)  27  6)  s 
28°33  26°S^ 


604*60ti 


29.6 


29°3 


59  S%  S9  is  59  10  5&4b  58  20 


57  5*167  28  57 


39302  3025 


3050  31  14314032 


3259 


**> 

428  Si 


G048  602(  60*4  5941 


29*i 


59  18  58  S4  Stf 


2934  295630i9  30 


58S 
3  Tad  3 1*55 


57^39 
32*21 


57  is  1 56  46  56  19 

32*47!  33*14  aiV 


39 


60°3I 


6010 


59*48  59*25  59  z 


29  7  29  29  29  So  30  tt  30  35 


30°» 


58  39)  58%4)  57  so)  57  24 
32°>6 


56  58  56  34  56  6 


5537 


40 


60  57  6056  60 15  59°53 


29*3 


93Z  59io 


58*47  58  *  58  o 


29V 


2945  30  7 


30283050  31    13 


1 57  35  57jio 
32*25  32*50 


564*156  19  5552|5S2*|54S7 
35*3 


33*,6 


34*8  3 ' 


A  I 

41 


29*40 


5939  59JI7 
30*Zi  30*43* 


31  5 


57*45 
32*5 


57^ 
32*39  33"  3 


33  2»  3354-3421 


55, -t.  5444- 
34*4«U5PI6 


54-J6 
35*44 
53*37 


30e,i 


S9*a*|  59*3 1 58*40  58°ie  57;s5|57°32  57"8 
32*5t 


3141 


3228 


56*3)56  19 
33*17  33*4i' 


55  53  55  27 


PROVIDENCE,    R.    I. 


TABLE   2. — (Continued.) 

ANGLE  OF  EDGE. 
GEAR. 


5655545352515049484746454443 


42 


12 


77  54  77  42 


77*5 
I2°45 


77V 
I3*o' 


76°46|76°3o  76V  75  se 
13*36 


75*4 
14*19 


14*37 


743 
IS°57 


13 


76*w 


13*4 


13  18 


13  32 


76'l3 
13*47 


75°6e  7^46  75*2*  7S°e 


14    18 


74*5i' 
IS*9' 


74» 
15*28 


74 
15*47 


(67 


16  28 


7246 

17V 


14 


75*58  75*43 


14*4 


»4I7' 


75*28 
I4*3i 


75V  74*56  74*39  74"z 


15*4 


is'e 


IS°39 


16*87 


73*44173*25 
16*45 


17*39 


15 


75V 


7A44 

lS*lfa 


74V  74*12  73°S5 


8  72  S3  7239 


72*18  71 


7lV 


71*10  70  46  70V 


1531 


1742 


19  14 


\99 


16 


74-3 
15*57 


7347 

«6*.3 


73V  73°i» 
16V 


72*54  72°35  72 


71"  12 


1745 


te's 


70*49  TOV 
9V 


«9*3S  69V 
>9'59|2oV  20*51 


17 


737 


7249 


16  53  17 


. 
1729 


T2"\i 

17*47 


71*54  71 


70 52  7030  70°7 


18*6 


1826 


«930 


1953 


6943  69  17 
2O*I7  20*43 


68*5268  0,67*5* 


18 


»749 


71  53 

18*7 


71*34 


7l°is 

I8°4S 


70*54 
19*6 


70*33 
19*27 


70*2 


69'so 


69V  69  3 


6838  68  iz  67 V  67  17 


203* 


2057  21  »  2148 


22 


2243 


19 


7lwis 
18*4S 


7057 

19°  I 


70*17 


69  34  69' 


684868 


1943  20 


20  «  2048 


2135 


67  S9  67  34  67  6 
22*2^  22*54 


663866  10 
23*22  23*50 


65*39 
24*2,' 


20 


70*2 
19V 


69*4 
20*.9 


69  19  68*57 


67  4d  67V  6657  66 V  6C2 


21   25  2148 


22  V 


23°  3 


23 


23V 


65*33  65*3 


64*32 

25 


21 


69V  69* 
2OV  20V 


67°37  67*  3  66*4B  66*«  6S°si  65*t8  64*59  64*»  63*S8 


22  2S 


2247  23  12 


2338 


24  S 


2432 


25*  • 


25*3 


26 


63*26 
26*34 


22  S 


68  33  68  12 


67  50  67  27  67°4 


66*40 


S  6549  65 13  64  55 


22°io  22°33  22*56  23*20 


24J7 


25*5 


64°t6  63*57 

25V  26*3 


63* »  62*54 
26*3* 


27*6 


62*21 
27V 


67*18 

22*42 


23°  S 


65*44  65*18  64*51 
24*16  24*42  25*9 


64*34  63  55  63  26  62*56 
25*36  26*  s  26V  27*A 


62V  6l*St  61* 


23*12  23  V 


6S°|4  64*48  64*22 


63V  63V  62*57  62*27  6l*s«  61*2)  60 V  60is 


2358  2422 


25  12 


2538  266 


263* 


27*3 


65*57  6533 


243 


2*27 


65*9 
24*5.' 


25*,5 


63*53(63*161  6ZVJ62V  61*59  61*29  6O*S7 
28%i  39*3' 


26*34  27*2 


273l 


28 


29  )•  30  io 


G5  6  6442  64*ie  63*52  63* 


62*»  62*34 


27*  . 


62*3 


60  31  59V  59*B  58  so 


27  29  27  57 


28  57  29  «  30  i 


30V  3l'io 


se 

3I*4« 


?Z 

28 
29 

30 

3] 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 


64' ib  63V 
25*4426*9' 


62*34 


61*3 


61*  •' 


60V  60*7 


26  3*  27  0 


27  26  27  50  28  22 


2«V  29°ti  29*! 


S9V  59*a 

soV  3oV 


5e*«i     _. 

»l*32  32*7 


S7i4 


63*«6  63*1 


62*3*  62*9 


26*34  26*S9  27V  27*si 


61*42 

28*8 


6045  60*i5  59°4S  59*i3  SCT4t  58*7 


57° »  S6*W 


28*46  29*15  29*45  30°i5 


3047  3I*2«  31*53  32*»  33*4 


33V 


62V  62*iz  61*45  6) 
27*23  27*48  28*15 


29*9 


60*23  59  53|  59  23  58  52  58*»  57*46  57ei2  56*37  56*» 
SI*  6  3l*4i  32°»4:  32*48 


29*37  30*7 


3037 


33°«  34*o 


5523 
34*»7 


5T27  se'ss 


28)1 


28  37  29  3 


293.' 


29  99  30  ta  30  s*  31 V  32o 


32» 


33*7 


33*4,' 


*C28 
34*C5  35*32 


28*l»  29*24  29V  30*19  50 V  3 


59  ia  &8 4t  58  >2  57*. 


3lV  32* »'  32° 


57*6 


•r«sci 


5526  54V  54  it 


53V 


33*24  33°59  34*34  35*l«  »5*48  36*26 


60  is  S9*48  59*i 


29*4530", 


30*3*3 


56V  56 19  55  *s  55' 


B4 


H  32  6 


3^37  33  8 


33  41  34  IS 


34*4935*2536 


S3*M  53*2 


36V  37°ie 


592959*2 


58*34  58*5 


30*31  30*58  31*26  31*55  3lV 


57  36  57  6  56°3»  56*2 


32  54  33  26 


33V 


55*30  54*56  54*2 


5345  53  8 


52*29  5I°SO 


3430 


3SV 


35  39  36  is  36V  37  3 


38' 


5844  58« 

3»*I6  3»*4* 


3^0 


9  5547  55'is  54* 
34*13  34*45  35*19 


54°V 


53V  52V  5Z»  5» 


3S  S3  36  28  37  8 


51*  »' 
3»o' 


58*0 
32  o 


S7*3i  57 


32283257  3327 


3357 


5S°o 
35*0 


5428  53  54  53  20 
36*40 


52*44 
37*16 


37°E7 


5l°3o  50  §i 
38*30  39*9 


39*48 


57  it  5648  56*19  5549  55*ia  54  47  54  IB  53V  53*8 


32  44  33  w  3341 


35*45  36*18  36*52  37*27  38*3 


52 33  BIS?  Bl*20  50*43 


38  4o'  39  >7 


49V 
39*56  40*3^ 


56  3t  56*4.' 


55  M  35  s 


33  EB  33  56  34  2S  34  sb  35  26 


35° 


36°3i  37  A 


37  37  38  13  38  48  39  2S 


v.  50°ss  49  s*  49 
40V 


4043 


41*23 


55si  55«i  54*52  5423  53°s 


35*8 


35*37  36*9 


51*3 
3»  42|  37l4|  37481 38*22 


50*27  49°4» 

38"  57  39*33-40' 


49*»i 


«  42  * 


559 

3451 


3S°2i' 


53°39  53*7 


52*3 


52*3 
r2i  36'53  37  «4  37*57  38"3i 


5l°t»  50°54  50*19  43°4a  49°S 


396 


39V,' 


47  48  47*?' 


40* 


3S°32  3* 


3734386 


3840 


39)4  3948402441 


«°SS  43°3* 


5348J53I7 
36*12 


52)6 
36°43  37*2  37*44 


5l*4s'Sl  it  SO  39  50  s' 


49  30  48  54  48' 


47*40  47* 


46  22  45V 


385 


38483921 


3955403041   f, 


41 43  42  20  42  59  43  »  44 ,9 


52*3*  52*8 

'•n  37*5Tj38'24|38'«6|39'29|40'i 


47*36  46*5S 
42*z4  43*  i' 


46*20 


41/40 


4340  44-20 


26 


BROWN    &    SHARPE    MFG.    CO, 


TABLE 


ANGLE  OF  FACE. 
GEAR. 


1140393837363534  333231 


30292827 


12 


1  3'37113-s 

70'ai 


70' 


14"  IB  14*39 
70*6'  69^ 


15 
69*s 


15  £4-  15  49 

67*4 


16 
67is 


1927  £0  5' 

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13 


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67*43 


16  51 

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» 

66*33 


65°,  6 


62*i 


14 


6  /5JI6  a 
>°3.  68*0 


16*59   17*4.4 

67*2966*5 


65*8 


9 
64*30 


63*6 


2066 
62*20 


22*,.T 
60*41 


15 


67*.  66 Va 


14654 


19  n 
65°3 


20   I 
63V 


23  10 
60*2 


23  S 

59*JS8°3|S7'4. 


16 


18  V«  19° 


I9'35  ZO0* 

64*5i  64*2i 


21  ' 
63*V 


iS\ 
57*< 


17 


205<  2lV 
63*«4  63% 


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22  « 
6lVo 


2353 
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24  10 


275s 


18 


>|92I\7 


6035 


24 ,« 

S9°5l 


24  «« 


2657 
56*39 


28z 
5449J5350I5247 


19 


[28  to]  22  4 
>i"*t\e>l'\ 


352 
60*4<4 


24t 


25°i 


2537 

58*37 


4261 


2738  28 »« 


574 


S+°, 


29561304: 
53°2f 


51*14 


,23*aJ24° 

'6I*3060<5J 


125*6 
59°^ 


Ukt 


26°55 
5725 


2.71 


28  15  28*58 

5549 


2944 


Si"*. 


5458  54* 


33 ' 
50* 


21 


s  59°4« 


26  is 

58*26 


26  53 

574* 


28*<o 
552. 


30*7 


31  °4- 


54°3«  53«a  SI 


32<>*»|33atj34*a 
5O*5J 


I254«26°9 


2653 


272 


28*« 
56*a< 


28 

6551 


SM  23 ° 


30  s 
54*7 


49VJ 


2836 


55*3 


29*4 35  a 
544- 


an 

53*6 


•  32' 


33°3i  34-2? 


36,*l37^t 

4-ft-'5o|470i7H>-62 


275728*3 


50  ° 


34"i 
49*o|49l!i 


36° 
47!, 


rS 

56 


523*151 


35  °, 
5pM49j 


36*5£ 

7 


3  74-H  3  84  j|  3  941 
46*i5 


30° ,  \30*s 


53°e 


35'a, 
49% 


38  46  39*55  40  SI 

45<>,oU46r|43 


3I°3   31  39 
t/    54°J53°3 


32V) 


3331 


35°3 


1549 


50  34  4  94  s  48*s$  4^ 


37Ss 
47% 


38  , 
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404  41    i 


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IssWstM 


335 
51°,, 


r. 

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3338 

51 


34 n 


3539 


4750 


4f,d 


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3  5  56 


38°7 


31 


|34"S3|35°3 

1504- (149  57 


36 

49°i3  48°t« 


3735 

4739 


3  6 Si 

46*3 


74S0, 


3941 
45% 


403Z 


41  24 

45*,7 


42l« 


4t*M 


1 354*  36*2  7 
»50  49°7 


376 


4049  4138 


36«937e 


38eo|38C4t 


46*46 


4558 


40  0 
45 


45%. 

40?; 


42  2e 


3752  38" t 
48*2  47  z  7 


38°sj 


40"le 
45°8 


41 
44t«| 


4149 

4319 


47lJ46°a9 


4554  45°a 


424i 


ifa 


43*34 


|4S52J45*8 


4248 


J40°47|4/0>a 
145% 


Jk  ID  I   .     u 
^4142" 


41 


PROVIDENCE,    R.    I. 


TABLE    $.— (Continued.) 

ANGLE  OF  FACE. 
GEAR. 


=  90"  -  (exb  + 
(See  page  13.) 


28 


TABLE  4 
ANGLE  OF  FACE. — GEAR. 


V 54,  9*  Z  \9'  9  \y  18  9*26  9° 3$  3 

17  si  77*42  rfmtrim  if  a' 


,,     . .  "  56  12  7 

IS  31  75  20  75  7  74  M  74447427 


TABLE  4.— (Continued.) 
ANGLE  OF   FACE. —  GEAR. 


29 


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43*46|43  4 

BROWN    &    SHARPE   MFG.    CO. 


NATUKAL  SINE. 


Deg. 

0' 

10' 

20' 

SO' 

40' 

50' 

60' 

0 

.00000 

.00291 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02326 

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.03199 

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i  88 

2 

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87 

3 

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.06975 

86 

4 

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.08135 

.08425 

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85 

5 

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.09005 

.09295 

.09584 

.09874 

.10163 

.10452 

84 

6 

.10452 

.10742 

.11031 

.11320 

.11609 

.11898 

.12186 

83 

7 

.12186 

.12475 

.12764 

.  13052 

.  13341 

.  13629 

.13917 

i  82 

8 

.13917 

.14205 

.14493 

.  14780 

.15068 

.  15356 

.15643 

1  81 

9 

.15643 

.15930 

.16217 

.16504 

.16791 

.17078 

.17364 

i  80 

10 

.17364 

.17651 

.17937 

.18223 

.18509 

.18795 

.19080 

!  79 

11 

.19080 

.19366 

.19651 

.19936 

.20221 

.20506 

.20791 

!  78 

12 

.20791 

.21075 

.21359 

.21644 

.21927 

.22211 

.22495 

77 

13 

.22495 

.22778 

.23061 

.23344 

.23627 

.23909 

.24192 

!  76 

14 

.24192 

.24474 

.24756 

.25038 

.25319 

.25600 

.25881 

75 

15 

.25881 

.26162 

.26443 

.26723 

.27004 

.27284 

.27563 

:  74 

16 

.27563 

.27843 

.28122 

.28401 

.28680 

.28958 

.29237 

i  73 

17 

.29237 

.29515 

.29793 

.30070 

.30347 

.30624 

.30901 

72 

18 

.30901 

.31178 

.31454 

.31730 

.32006 

.32281 

.32556 

71 

19 

.32556 

.32831 

.33106 

.33380 

.33654 

.33928 

.34202 

1  70 

20 

.34202 

.34475 

.34748 

.35020 

35293 

.35565 

.35836 

69 

21 

.35836 

.36108 

.36379 

.36650 

.36920 

.37190 

.37460 

!  68 

22 

.37460 

.37730 

.37999 

.38268 

.38536 

.38805 

.39073 

67 

23 

.39073 

.39340 

.39607 

.39874 

.40141 

.40407 

.40673 

I  66 

24 

.40673 

.40939 

.41204 

.41469 

.41733 

.41998 

.42261 

i  65 

25 

.42261 

.42525 

.42788 

.43051 

.43313 

.43575 

.43837 

!  64 

26 

.43837 

.44098 

.44359 

.44619 

.44879 

.45139 

.45399 

|  63 

27 

.45399 

.45658 

.45916 

.46174 

.46432 

.46690 

.46947 

62 

28 

.46947 

.47203 

.47460 

.47715 

.47971 

.48226 

.48481 

61 

29 

.48481 

.48735 

.48989 

.49242 

.49495 

.49747 

.50000 

!  60 

30 

.50000 

.50251 

.50503 

.50753 

.51004 

.51254 

.51503 

59 

31 

.51503 

.51752 

.52001 

.52249 

.52497 

.52745 

.52991 

58 

32 

.52991 

.53238 

.53484 

.53730 

.53975 

.54219 

.54463 

57 

33 

.54463 

.54707 

.54950 

.55193 

.  55436 

.55677 

.55919 

56 

34 

.55919 

.56160 

.56400 

.56640 

.56880 

.57119 

.57357 

55 

35 

.57357 

.57595 

.57833 

.58070 

.58306 

.58542 

.58778 

54 

36 

.58778 

.59013 

.59248 

.59482 

.59715 

.59948 

.60181 

53 

37 

.60181 

.60413 

.60645 

.60876 

.61106 

.61336 

.61566 

52 

38 

.61566 

.61795 

.62023 

.62251 

.62478 

.62705 

.62932 

51 

39 

.62932 

.63157 

.63383 

.63607 

.63832 

.64055 

.64278 

.  50 

40 

.64278 

.64501 

.64723 

.64944 

.65165 

.65386 

.65605 

i  49 

41 

.65605 

.65825 

.66043 

.66262 

.66479 

.66696 

.66913 

48 

42 

.66913 

.67128 

.67344 

.67559 

.67773 

.67986 

.68199 

47 

43 

.68199 

.68412 

.68624 

68835 

.69046 

.69256 

.69465 

46 

44 

.69465 

.69674 

.69883 

.70090 

.70298 

.70504 

.70710  |  45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

r»e£. 

NATUBAL   COSINE. 


PROVIDENCE,    R.    I. 


NATUBAL   SINE. 


Deg. 

0'       10' 

20' 

30' 

40' 

50' 

w 

45 

.70710 

.70916 

.71120 

.71325 

.71528 

.71731 

.71934 

44 

46 

1  .71934 

.72135 

.72336 

.72537 

.72737 

.72936 

.73135 

43 

47 

.73135 

.73333 

73530 

.73727 

.73923 

.74119 

.74314 

42 

48 

.74314 

.74508 

.74702 

.74895 

.75088 

.75279 

.75471 

41 

49 

.75471 

.75661 

.75851 

.76040 

.76229 

.76417 

.76604 

40 

50 

.76004 

.76791 

.76977 

.77102 

.77347 

.77531 

.77714 

39 

51 

.77714 

.77897 

.78079 

.78260 

.78441 

.78621 

.78801 

1  38 

53 

.78801 

.78979 

.79157 

.79335 

.79512 

.79688 

.79863 

37 

53 

.79863 

.80038 

.80212 

.80385 

.80558 

.80730 

.80901 

1  36 

54 

.80901- 

.81072 

.81242 

.81411 

.81580 

.81748 

.81915 

35 

55 

.81915 

.82081 

.82247 

.82412 

.82577 

.82740 

.82903 

34 

56 

.82903 

.83066 

.83227 

.83388 

.83548 

.83708 

.83867 

33 

57 

.83867 

.84025 

.84182 

.84339 

.84495 

.84650 

.84804 

32 

58 

.84804 

.84958 

.85111 

.85264 

.85415 

.85566 

.85716 

31 

59 

.85716 

.85866 

.86014 

.86162 

.86310 

.86456 

.86602 

30 

60 

.86602 

.86747 

.86892 

.87035 

.87178 

.87320 

.87462 

29 

61 

.87462 

.87602 

.87742 

.87881 

.88020 

.88157 

.88294 

28 

62 

.88294 

.88430 

.88566 

.88701 

.88835 

.88968 

.89100 

27 

63 

.89100 

.89232 

.89363 

.89493 

.89622 

.89751 

.89879 

26 

64 

.89879 

.90006 

.90132 

.90258 

.90383 

.90507 

.90630 

25 

65 

.90630 

.90753 

.90875 

.90996 

.91116 

.91235 

.91354 

24 

66 

.91354 

.91472 

.91589 

.91706 

.91821 

.91936 

.92050 

23 

67 

.92050 

.92163 

.92276 

.92388 

.92498 

.92609 

.92718 

22 

68 

.92718 

.92827 

.92934 

.93041 

.93148 

.93253 

.93358 

21 

69 

.93358 

.93461 

.93565 

.93667 

.93768 

.93869 

.93969 

20 

70 

.93969 

.94068 

.94166 

.94264 

.94360 

.94456 

.94551 

19 

71 

.94551 

.94646 

.94789 

.94832 

.94924 

.95015 

.95105 

18 

'72 

.95105 

.95195 

.95283  • 

.95371 

.95458 

.95545 

.95630 

17 

73 

.95630 

.95715 

.95799 

.95882 

.95964 

.96045 

.96126 

16 

74 

.96126 

.96205 

.  96284 

.96363 

.96440 

.96516 

.96592 

15 

75 

.96592 

.96667 

.96741 

.96814 

.96887 

.96958 

.97029 

14 

76 

.97029 

.97099 

.97168 

.97237 

.97304 

.97371 

.97437 

13 

77 

.97437 

.97502 

.97566 

.9762!) 

.97692 

.97753 

.97814 

12 

78 

.97814 

.97874 

.97934 

.97992 

.98050 

.98106 

.98162 

11 

79 

.98162 

.98217 

.  98272 

.98325 

.98378 

.98429 

.98480 

10 

80 

.98480 

.98530 

.98580 

.98628 

.98676 

.98722 

.98768 

9 

81 

.98768 

.98813 

.98858 

.98901 

.98944 

.98985 

.99026 

8 

82 

.99026 

.99066 

.09106 

.99144 

.99182 

.99218 

.99254 

7 

83 

.99254 

.99289 

.99323 

.99357 

.99389 

.99421 

.99452 

6 

84 

.99452 

.99482 

.99511 

.99539 

.99567 

.99593 

.99619 

5 

85 

.99619 

.99644 

.99668 

.99691 

.99714 

.99735 

.99756 

4 

86 

.99756 

.99776 

.99795 

99813 

.99830 

.99847 

.99863 

3 

87 

.99863 

.99877 

.99891 

.99904 

.99917 

.99928 

.99939 

2 

88 

.99939 

.99948 

.99957  !  .99965 

.99972 

.99979 

.99984 

1 

89 

.99984 

.99989 

.99993  |  .99996 

.99998 

.99999 

1.0000 

0 

- 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATUKAL   COSINE. 


BROWN  &  bHARPE  MFG.  CO. 


NATUKAL  TANGENT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

.00000 

.00290 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02327 

.02618 

.02909 

.03200 

.03492 

88 

2 

.03492 

.03783 

.04074 

.04366 

.04657 

.04949 

.05240 

87 

3 

.05240 

.05532 

.05824 

.06116 

.06408 

.06700 

.06992 

86 

4 

.06992 

.07285 

.07577 

.07870 

.08162 

.08455 

.08748 

85 

5 

.08748 

.09042 

.09335 

.09628 

.09922 

.10216 

.10510 

84 

6 

.10510 

.  10804 

.11099 

.11393 

.11688 

.11983 

.12278 

83 

7 

.12278 

.12573 

.12869 

.13165 

.13461 

.  13757 

.14054 

!  82 

8 

.14054 

.14350 

.14647 

.  14945 

.15242 

.15540 

.15838 

81 

9 

.15838 

.16136 

.16435 

.16734 

.17033 

.17332 

M7632 

:  80 

10 

.17632 

.17932 

.18233 

.18533 

.18834 

.19136 

.  19438 

79 

11 

.19438 

.  19740 

.20042 

.20345 

.20648 

.20951 

.21255 

78 

12 

.21255 

.21559 

.21864 

.22169 

.22474 

.22780 

.23086 

77 

13 

.23086 

.23393 

.23700 

.24007 

.24315 

.24624 

.24932 

76 

14 

.24932 

.25242 

.25551 

.25861 

.26172 

.26483 

.26794 

75 

15 

.26794 

.27106 

.27419 

.27732 

.28046 

.28360 

.28674 

74 

16 

.28674 

.28989 

.29305 

.29621 

.29938 

.30255 

.30573 

73 

17 

.30573 

.30891 

.31210 

.31529 

.31850 

32170 

.  32492 

72 

18 

.32492 

.32813 

.33136 

.33459 

.33783 

.34107 

.34432 

71 

19 

.34432 

.34758 

.35084 

.35411 

.35739 

.36067 

.36397 

70 

20 

.36397 

.36726 

.37057 

.37388 

.37720 

.38053 

.38386 

69 

21 

.38386 

.38720 

.39055 

.39391 

.39727 

.40064 

.40402 

68 

22 

.40402 

.40741 

.41080 

.41421 

.41762 

.42104 

.42447 

67 

23 

.42447 

.42791 

.43135 

.43481 

.43827 

.44174 

.44522 

66 

24 

.44522 

.44871 

.45221 

.45572 

.45924 

.46277 

.46630 

65 

25 

.46630 

.46985 

.47341 

.47697 

.48055 

.48413 

.48773 

64 

26 

.48773 

.49133 

.49495 

.49858 

.50221 

.50586 

.50952 

63 

27 

.50952 

.51319 

.51687 

.52056 

.52427 

.52798 

.53170 

62 

28 

.53170 

.53544 

.53919 

.54295 

.54672 

.55051 

.55430 

61 

29 

.55430 

.55811 

.56193 

.56577 

.56961 

.57847 

.57735 

60 

30 

.57735 

.58123 

.58513 

.58904 

.59297 

.59690 

.60086 

59 

31 

.60086 

.60482 

.60880 

.61280 

.61680 

.62083 

.62486 

58 

32 

.62486 

.62892 

.63298 

.63707 

.64116 

.64528 

.64940 

57 

33 

.64940 

.65355 

.65771 

.66188 

.66607 

.67028 

.67450 

56 

34 

.67450 

.67874 

.68300 

.68728 

.69157 

.69588 

.70020 

55 

35 

.70020 

.70455 

.70891 

.71329 

.71769 

.72210 

.72654 

54 

36 

.72654 

.73099 

.73546 

.73996 

.74447 

.74900 

.75355 

53 

37 

.75355 

.75812 

.76271 

.76732 

.77195 

.77661 

.78128 

52 

38 

.78128 

.78598 

.79069 

.79543 

.80019 

.80497 

80978 

51 

39 

.80978 

.81461 

.81946 

.82433 

.82923 

.83415 

.83910 

50 

.40 

.83910 

.84406 

.84906 

.85408 

.85912 

.86419 

.  86928 

49 

41 

.86928 

.87440 

.  87955 

.88472 

.88992 

.89515 

.90040 

48 

42 

.90040 

.90568 

.91099 

.91633 

.92169 

.92709 

.93S51 

47 

43 

.93251 

.93796 

.94345 

.94896 

.95450 

.96008 

.96568 

46 

44 

.96568 

.97132 

.97699 

.98269 

.98843 

.99419 

1.0000 

45 

- 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


PROVIDENCE,    R.    I, 


33 


NATURAL  TANGENT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

* 

60 

45 

1.0000 

1.0058 

1.0117 

1.0176 

1.0235 

1  0295 

1.0355 

44 

46 

1.0355 

1.0415 

1.0476 

.0537 

1.0599 

1.0661 

1.0723 

,43 

47 

1.0723 

1.0786 

1.0849 

.0913 

1.0977 

1.1041 

1.1106 

42 

48 

1.1106 

1.1171 

1.1236 

.1302 

1.1369 

1.1436 

1.1503 

41 

49 

1.1503 

1.1571 

1.1639 

.1708 

1.1777 

1.1847 

1.1917 

40 

50 

1.1917 

1.1988 

1.2059 

.2131 

1.2203 

1.2275 

1  2349 

39 

51 

1.2349 

1.2422 

1.2496 

.2571 

1.2647 

1.2723 

.2799 

38 

52 

1.2799 

1.2876 

1.2954 

.3032 

1.3111 

1.3190 

.3270 

37 

53 

1.3270 

1.3351 

1.3432 

1.3514 

1.3596 

1.3680 

.3763 

36 

54 

1.3763 

1.3848 

1.3933 

1.4019 

1.4106 

1.4193 

.4281 

35 

55 

1.4281 

1.4370 

1.4459 

1.4550 

1.4641 

1.4733 

.4825 

34 

56 

1.4825 

1.4919 

1.5013 

1.5108 

1.5204 

1.5301 

.5398 

33 

57 

1.5398 

1.5497 

1.5596 

1.5696 

1.5798 

1.5900 

.6003 

32 

58 

1.6003 

1.6107 

1.6212 

1.6318 

1.6425 

1.6533 

1.6642 

31 

59 

1.6642 

1.6753 

1.6864 

1.6976 

1.7090 

1.7204 

1.7320 

30 

60 

1.7320 

1.7437 

1.7555 

1  .  7674 

1.7795 

1.7917 

1.8040 

29 

61 

1.8040 

1.8164 

1.8290 

1.8417 

1.8546 

1.8676 

1.8807 

28 

62 

1.8807 

1.8940 

1.9074 

1.9209 

1.9347 

1.9485 

1.9626 

27 

63 

1.9626 

1.9768 

1.9911 

2.0056 

2.0203 

2.0352 

2.0503 

26 

64 

2.0503 

2.0655 

2.0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

65 

2.1445 

2.1609 

2.1774 

2.1943 

2.2113 

2.2285 

2.2460 

24 

66 

2.2460 

2.2637 

2.2816 

2.2998 

2.3183 

2.3369 

2.3558 

23 

67 

2.3558 

2.3750 

2.3944 

2.4142 

2.4342 

2.4545 

2.4750 

22 

68 

2.4750 

2.4959 

2.5171 

2.5386 

2.5604 

2.5826 

2.6050 

21 

69 

2.6050 

2.6279 

2.6510 

2.6746 

2.6985 

2.7228 

2.7474 

20 

70 

2.7474 

2.7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 

71 

2.9042 

2.9318 

2.9600 

2.9886 

3.0178 

3.0474 

3.0776 

18 

72 

3.0776 

3.1084 

3.1397 

3.1715 

3.2040 

3.2371 

3.2708 

17 

73 

3.2708 

3.3052 

3.3402 

3.3759 

3.4123 

3.4495 

3.4874 

16 

74 

3.4874 

3.5260 

3.5655 

3.6058 

3.6470 

3.6890 

3.7320 

15 

75 

3.7320 

3.7759 

3.8208 

3.8667 

3.9136 

3.9616 

4.0107 

14 

76 

4.0107 

4.0610 

4.1125 

4.1653 

4.2193 

4.2747 

4.3314 

13 

77 

4.3814 

4.3896 

4.4494 

4.5107 

4.5736 

4.6382 

4.7046 

12 

78 

4.7046 

4.7728 

4.8430 

4.9151 

4.9894 

5.0653 

5.1445 

11 

79 

5.1445 

5.2256 

5.3092 

5.3955 

5.4845 

5.5763 

5.6712 

10 

80 

5.6712 

5.7693 

5.8708 

5.9757 

6.0844 

6.1970 

6.3137 

9 

81 

6.3137 

6.4348 

6.5605 

6.6911 

6.8269 

6.9682 

7.1153 

8 

82 

7.1153 

7.2687 

7.4287 

7.5957 

7.7703 

7.9530 

8.1443 

7 

83 

S.1443 

8.3449 

8.5555 

8.7768 

9.0098 

9.2553 

9.5143 

6 

84 

9.5143 

9.7881 

10.078 

10.385 

10.711 

11  059 

11.430 

5 

85 

11.430 

11.826 

12.250 

12.706 

13.196 

13  726 

14.300 

4 

86 

14.300 

14.924 

15.604 

16.349 

17.169 

18.075 

19.081 

3 

87 

19.081 

20.205 

21.470 

22.904 

24.541 

26.431 

28.636 

2 

88 

28.636 

31.241 

34.367 

38.188 

42.964 

49.103 

57.290 

1 

89 

57.290 

68.750 

85.939 

114.58 

171.88 

343.77 

00 

0 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


34 


BROWN    &   SHARPE    MFG.    CO. 


CHAPTER     IV. 

WORM  AND  WORM  WHEEL 

(Fig.  8.) 


PROVIDENCE,    R.    I.  35 


FORMULAS. 

L  =  lead  of  worm. 
N  =  number  of  teeth  in  gear. 
m  =  threads  or  turns  per  inch  in  worm, 
d=  diameter  of  worm. 
d'  =  diameter  of  hob. 
T  =  throat  diameter. 
B  =  blank  diameter  (to  sharp  corners). 
C  =  distance  between  centers. 
o  =  thickness  of  hob-slotting  cutter. 
/=  width  of  lands  at  bottom. 
b  =  pitch  circumference  of  worm. 
v  —  width  of  worm  thread  tool  at  end. 
w  =  width  of  worm  thread  at  top. 
P  =  diametral  pitch. 
P1  =  circular  pitch. 
.$•  =  addendum  and  module. 
/  =  thickness  of  tooth  at  pitch  line. 
tn  =  normal  thickness  of  tooth. 
/=  clearance  at  bottom  of  tooth. 
D"  =  working  depth  of  toolh. 
D"  +/  =  whole  depth  of  tooth. 

tf  =  angle  of  tooth  of  worm  wheel  with  its  axis,  or  the 
angle  of  thread  of  worm  with  a  line  at  right  angles  to  its  axis. 
If  the  lead  is  for  single,  double,  triple,  etc.,  thread,  then 
I,  =  P,  2  P',  3  P,  etc. 


36  BROWN    &   SHARPE   MFG.   CO. 


a  =  60°  to  9<DC 


m 


N    +    2 

D  =  NP/  =  N 

7T 


(</  _  2  j) 

_  L       j  Practical  only  when  width  of  wheel  on  wheel  pitch  circle 


«  _  L       j  Practi 
~fr        \          is 


~  not  more  than  j/3  pitch  diameter  of  worm. 


2 

TD  _  -p    ,         /!_      i  Of\        A  measurement  of  sketch  is  generally 

tf  -   1    +  2  I  r     -  r  COS  -I  sufficient. 


^  =  ^+   2/ 

*  =  .3iP' 

^-.335?' 


NOTE. — The  notations  and  formulas  referring  to  tooth  parts,  given  on  page  5  for 
spur  gears,  apply  to  worm  wheels,  and  are  here  used. 

NOTE. — Hob  and  worm  should  be  marked,  as  per  example  : 
4  turns  per  i''  single  .25  P';  .25  L. 
2  turns  per  i"  double  .25  P';  .50  L. 


PROVIDENCE,   R.  I. 


37 


UNDERCUT  IN  WORM  WHEELS. 

In  worm  wheels  of  less  than  30  teeth  the  thread  of  the  worm 
(being  29°)  interferes  with  the  flank  of  the  gear  tooth.  Such 
a  wheel  finished  with  a  hob  will  have  its  teeth  undercut.  To 
avoid  this  interference  two  methods  may  be  employed. 

First  Method.  —  Make  throat  diameter  of  wheel 


cos 


—  +  4.$- 


or 


T  =  -937  N 

P 

This  formula  increases  the  throat  diameter,  and  conse- 
quently the  center  distance.  The  amount  of  the  increase  can 
be  found  by  comparing  this  value  of  T  with  the  one  as  obtained 
by  formula  on  page  36.  To  keep  the  original  center  distance, 
the  outside  diameter  of  the  worm  must  be  reduced  by  the 
same  amount  the  throat  diameter  is  increased. 

Second  Method. — Without  changing  any  of  the  dimensions 
we  found  by  the  formulas  given  on  page  36,  we  can  avoid  the 
interference  to  be  found  in  worm  wheels  of  less  than  30  teeth 
by  simply  increasing  the  angle  of  worm  thread.  We  find  the 
value  of  this  angle  by  the  following  formula  : 
Let  there  be 

2  y  =  angle  o±  worm  tnreaa. 

N  =  number  of  teeth  in  worm  wheel. 

cos  y 


From  this  formula  we  obtain  the  following  values  : 


N 

29 

28 

27 

26 

25 

24 

23 

22 

21 

2  y 

3oK 

3i 

$ 

32^ 

32^ 

33^ 

34K 

35 

36 

N 

19 

18 

17 

16 

15 

14 

13 

12 

2  y 

38 

39 

40 

4Iy2 

42% 

44/2 

46^ 

48 

20 


37 


As  this  latter  formula  involves  the  making  of  new  hobs  in 
many  cases,  on  account  of  change  of  angle,  we  prefer  to  reduce 
the  diameter  of  worm  as  indicated  by  first  method,  if  the  dis- 
tance of  centers  must  be  absolute. 


BROWN    &    SHARPS    MFG.    CO. 


SO 


99 


CD 


TURNS 
PER  INCH 


HOlId 


PROVIDENCE,    11.    I. 


39 


4, 


fcl 


o? 


CO 


CO 


CD 


1 


H* 


CO 


HOiid 


BROWN  &   SHARPK  MFG.   CO. 


CHAPTER  v. 


SPIRAL  OR  SCREW  GEARING, 

(Figs.  9,  10,  n.) 


RIGHT    HAND    SPIRAL    GEARS. 

In  spiral  gearing  the  wheels  have  cylindrical  pitch  surfaces, 
but  the  teeth  are  not  parallel  to  the  axis.  The  line  in  which 
the  pitch  surface  intersects  the  face  of  a  tooth  is  part  of  a 
screw  line,  or  helix,  drawn  at  the  pitch  surface.  A  screw 
wheel  may  have  one  or  any  number  of  teeth.  A  one-toothed 
wheel  corresponds  to  a  one-threaded  screw,  a  many-toothed 
wheel  to  a  many- threaded  screw.  The  axes  may  be  placed  at 
any  angle. 

Consider  spiral  gears  with  : 

I.  Axes  parallel. 
II.  Axes  at  right  angles. 
III.  Axes  any  angle. 


PROVIDENCE,   R.  I. 


"  S-  number  of  teeth  in  gears  •!   * 


Fig.  10. 

LEFT    HAND    SPIRAL    GEAR. 

Let  there  be : 

N,= 

N, 

C  =  center  distance. 

P  =  diametral  pitch 

P'  =  circular  pitch. 
Pn  =  normal  diametral  pitch. 
P'n  =  normal  circular  pitch. 

y  =  angle  of  axes. 

Lj  =  exact  lead  of  spiral  on  pitch  surface. 
LQ  =  approximate  lead  of  spiral  on  pitch  surface. 

T  =  number  of  teeth  marked  on  cutter  to  be  used  when 
teeth  are  to  be  cut  on  milling  machine. 

D  =  pitch  diameter. 

B  =  blank  diameter. 

^a  =   [  angle  of  teeth  with  axis 

/  =  thickness  of  tooth. 
s  =  addendum  and  module. 
D"  +  /  =  whole  depth  of  tooth. 

NOTE. — Letters  a  and  b  occurring  at  bottom  of  notations  refer  to  gears  a  and  b. 

I. — AXES  PARALLEL. 

Gears  of  this  class  are  called  twisted  gears.  The  angle  of 
teeth  with  axes  in  both  gears  must  be  equal  and  the  spirals 
run  in  opposite  directions.  The  angles  are  generally  chosen 
small  (seldom  over  20°)  to  avoid  excessive  end  thrust.  End 
thrust  may,  however,  be  entirely  avoided  by  combining  two 
pairs  of  wheels  with  right  and  left-hand  obliquity.  Gears  of 
this  class  are  known  as  Herringbone  gears.  They  are  com- 
paratively noiseless  running  at  high  speed. 


42  BROWN  &  SHARPK   MFG.  CO. 

II.  —  AXES  AT  RIGHT  ANGLES. 
Here  we  must  always  have  : 

1.  The  teeth  of  same  hand  spiral  ; 

2.  The  normal  pitches  equal  in  both  gears  ;  and 

3.  The  sum  of  the  angles  of  teeth  with  axes  =  90°. 

CHOOSING  ANGLE  OF  TEETH  WITH  AXES. 

1.  If  in  a  pair  of  gears  the  ratio  of  the  number  of  teeth  is 
equal  to  the  direct  ratio  of  the  diameters,  /.  *.,  if  the  number  of 
teeth  in  the  two  gears  are  to  each  other  as  their  pitch  diame- 
ters, then  the  angles  of  the  spirals  will  be  45°  and  45°  ;  for,  this 
condition  being  fulfilled,  the  circular  pitches  of  the  two  gears 
must  be  alike,  which  is  only  possible  with  angles  of  45°.     In 
such  a  combination  either  gear  may  be  the  driver. 

2.  If  the  ratio  of  the  diameters  determined  upon  is  larger 
or  smaller  than  the  ratio  of  the  number  of  teeth,  then  the 
angles  are  : 


In  such  gears  the  velocity  ratio  is  measured  by  the  number 
of  teeth,  and  not  by  the  diameters. 
3.  Given  Na,  N&  and  C  : 
If  P0'  is  made  =  P6',  then  we  have  case  "  i  "  and 

p,  _  7t  C 


But  if  Pa'  is  assumed,  then  : 


and 

tan  aa  =  ^         tan  ab  =  ?1 

"  b  *  a 

The  gear  whose  P'  or  a  is  larger  will  ordinarily  be  the 
driver,  on  account  of  the  greater  obliquity  of  the  teeth. 

4.  Given  N0,  N6  and  C  or  D. 
See  case  "  7  "  under  III.,  considering  ;/  =  90°. 

III. — Axis  AT  ANY  ANGLE  (y}. 

5.  Given  case  "  i,"  under  II.,  then  angles  of  spirals  =  }4  ^, 
for  the  same  reason. 

6.  Analogous  cases  to  "2"  and  "3,"  under  II.,   may   be 
worked  out,  when  angles  of  axes  =  7,  but   they   have   been 


PROVIDENCE,   R.  I. 


43 


omitted,  partly  because  the  formulas  are  too  cumbersome,  and 
partly  because  they  are  to  some  extent  covered  by  cases  "5  " 
and  'k7." 

7.  Given  Na,  N6  and  C,  or  one  of  the  pitch  diameters.  We 
find  the  angles  by  a  graphic  method,  which  for  all  practical 
purposes  is  accurate  enough  ;  ro  and  v  o  are  the  axes  of  gears 
forming  angle  y  (see  diagram,  Fig.  n.)  On  these  axes  we 
lay  off  lines  o  r  and  o  v  representing  the  ratio  of  the  number 
of  teeth  (velocity  ratio),  so  that  Na  :  N6  :  :  r  s  :  s  v,  and 


Fig.  11. 


construct  parallelogram  o  r  s  v.  Then,  according  to  Mc- 
Cord,*  the  angles  formed  by  the  tangent  s  o  in  the  pitch  con- 
tact o  with  the  axes  of  the  gears  insures  the  least  amount  of 
sliding.  In  bisecting  angle  y  by  tangent  u  o  and  using  angles 
produced  in  this  manner  we  equally  distribute  the  end  thrust  on 
both  shafts.  Both  methods  have  their  advantages  ;  to  profit 
by  .both  we  select  angles  cxa  and  abt  produced  by  tangent  o  x, 
bisecting  angle  u  o  s. 

Thus  we  have  when  angles  are  found  and  C  given, 
„/„         2  C  n  cos  an  cos  al 


and  when  Da  given 


Nacos 


Da  7t  COS  (Xa 

~N7 


and 


7t  cos  ab 


*  McCord,  Kinematics,  page  278. 


44  BROWN  &  SHARPE  MFG.  CO. 

GENERAL  FORMULAS, 

y  ~  aa  +  ab 
Patn  =  Pb'n 


TTCOSfX 


B  =  D  +  2  .r      or  =  D  +  -. 


or= 


N  cos  a 

p'n  _  p'  COg  a 


Pn=  JL     (Pitch  of  cutter.) 


pm 


D"+/  =  2  J  -f 

10 

T  =  -5L-  (5^  Note  7.) 

cos3** 

NP'  NTT 

°r 


tan  a  P  tan  a  [  L,i&  =  N&  P'a 

La  =  I0  W  ^«  (6"^  A^^/^  2  and  examples^) 

S  Gt 

cos   45°=    70711 
cos3  45°=    .3535 
tan  45°  =  i.ooo 

NOTE  i. — Cutters  of  regular  involute  system. 


Use  No.  i  cutter  for  T  from      135  up. 

«       2     i.        .«   «     «  55  to  134 


26  to    34 


No.  5  cutter  for  T  from  21  to  25 


6       ''        "     "      "  i?  to  20 

y  i*  «'  "  ** 


35  to    54       "     •»      J4  to  16 


12  to  13 


Note  2.— Gears  used  on  spiral  head  and  bed   for   Brown   &   Sharpe  milling 

machine : 

W  =  number  of  teeth  in      gear  on  worm. 

Gi  =  "  "      ist   "          stud. 

G2  =  "  '*      2d    "          stud. 

S  =  "  *'  '.'          screw. 

Should  a  spiral  head  of  different  construction  be  used,  the  formula  might  not 
apply. 


PROVIDENCE,    R.  I. 


45 


The  following  data  are  usually  required  in  cutting  spiral 
gears  in  a  Universal  Milling  Machine,  and  it  will  be  found 
convenient  to  arrange  them  in  tabular  form  as  follows  : 


GEAR. 


PINION. 


No.  of  Teeth     -     - 

Pitch  Diameter  - 

Outside  Diameter 

Circular  Pitch    -    -     -     - 

Angle  of  Teeth  with  Axis 

Normal  Circular  Pitch 

Pitch  of  Cutter  -     - 

Addendum  s 

Thickness  of  Tooth  t 
Whole  Depth  D"+f  - 

No.  of  Cutter 

Exact  Lead  of  Spiral  ------- 

Approximate  Lead  of  Spiral     -     -     -     - 

Gears  on  Milling  Machine  to  Cut  Spiral 

Gear  on  Worm 

ist  Gear  on  Stud    - 

2nd  Gear  on  Stud  -------- 

Gear  on  Screw  -     - 


If  the  exact  lead  L!  can  be  obtained  by  the  gears  at  hand, 
LI  will  equal  L2  and  we  shall  have  from  the  formula 
io  W  G2 

S  Gt 

L,       W  Go 
-J  =   Q-^r-     (for  B.  &  S.  Milling  Machine.) 

I O  o    ^--^i 

Example  I. 

Required  the  gears  for  cutting  a  spiral  of  2%"  lead. 

~-  =  -  factoring,  in  the  most  simple  way,  we  have 


XI        i  X  28        32  x  28       W  G2 


2X2         56  X  2         56  X  64 


46  BROWN  &  SHARPK   MFG.   CO. 

Thus  the  gearing  will  be  32  T.  on  worm,  64  T.  ist.  on  stud, 
28  T.  2nd  on  stud,  and  56  T.  on  screw. 

Trying  these  gears  on  the  Milling  Machine  we  find  that 
they  cannot  be  used,  and  as  we  have  no  other  regular  gears 
in  the  ratio  of  2  to  I  that  can  be  used  we  must  try,  by  factor- 
ing, to  get  such  ratios  for  the  two  pairs  of  gears  as  to  be  able 
to  use  the  gears  at  hand,  bearing  in  mind  that  the  combined 
ratio  must  be  J. 

i        18.      3x6       24  x  6       24  x  48 

4  =  72  =  9T8  ==  9  x  64  ==  72  x  64 

These  gears  are  at  hand  and  the  combination  can  be  used 
on  the  machine,  giving  the  exact  lead  of  2$". 

Example  II. 

Required  the  gears  for  cutting  a  spiral  of  8.639"  lea-d. 

8.639  =  8T60%9u- ;  reducing,  by  continued  fractions,  to  a 
smaller  fraction  of  approximately  the  same  value,  as  described 
on  pages  74  and  75 


639  )  1000  (  i 
639 

361  )639(  i 
361 

278  )  361  (  i 
278 

83  )  278  (  3 
249 

29  )  83  (  2 

58 

25 )  29  ( I 
25 

4)25(6 
24 

1)4(4 
4 


PROVIDENCE,   R.  I.  47 

1113216         4 


Selecting  ||  as  an  approximation  near  enough  for  our 
purpose,  and  in  fact  as  near  as  we  are  likely  to  find  gears  for, 
we  have  for  our  lead  8|f.  Applying  the  formula  as  in  Ex- 
ample I. 


_    W_G2 

10    "  ~   S   d 


216       108 
ijjjo  ==  7^5 

9  X  12        9  X48         72  X  48 


o        ijjjo  ==  7^5  factormg  we  have 


the 


these  being  regular  gears  furnished  with  the  Milling  Machine. 

Proof  : 

72  x  48  x  io 

-  =  0.040     =  .Lo 

100  x  40 

8.639  =L!  , 

.001"  error  in  lead. 

In  shops  where  much  work  is  done  in  milling  spirals  it  is 
desirable  to  have  a  full  set  of  gears  for  the  milling  machine, 
from  the  smallest  to  the  largest  numbers  of  teeth  that  can  be 
used.  This  makes  it  possible,  in  most  cases,  to  get  closer 
approximations  than  could  be  otherwise  obtained,  and  often 
saves  a  great  deal  of  figuring. 

When  the  use  of  continued  fractions  does  not  bring  a 
close  enough  approximation,  one  method  to  secure  a  closer 
result  is  to  add  to  or  substract  from  the  numerator  and  de- 
nominator of  the  fraction  to  be  reduced,  any  numbers  nearly 
in  proportion  to  the  given  fraction,  seeing  that  the  numbers 
added  or  substracted  are  such  as  to  make  the  fraction  reduc- 
ible to  lower  terms.  By  a  little  ingenuity  and  patience  ex- 
tremely close  approximations  can  generally  be  reached  in 
this  way. 

Take,  as  an  illustration,  the  fraction  in  Example  II. 


_ 

IO  IOOOO 

Adding  9  to  the  numerator  and  io  to  the  denominator,  these 


48  BROWN  &  SHARPK  MFG.   CO. 

being  in  about  the  same  ratio  to  each  other  as  the  numerator 
and  denominator  of  the  fraction,  we  have 

8639+9  =  8648     _  4324  _  47  x  92 
10000+10  =  10010       5005       55  x  91 

All  of  the  gears  in  this  case  are  special. 

Applying  the  same  proof  as  in  Example  II.  we  find  that 
this  train  of  gears  will  give  a  lead  of  8.6393+,  making  an 
error  of  .0003"  in  the  lead. 

No  doubt  a  much  closer  approximation  even  than  this 
could  be  obtained  by  further  trial. 

Another  method  is  to  multiply  both  terms  of  the  fraction 
by  some  number  which  will  make  one  term  of  the  fraction 
easily  reducible,  and  adding  one  to  or  subtracting  it  from  the 
other  term  to  make  it  possible  to  reduce  that  also. 

There  is  an  element  of  uncertainty  in  both  these  methods, 
as  we  never  feel  sure  that  we  have  obtained  the  best  combina- 
tion ;  practical  work,  however,  rarely  requires  accuracy  beyond 
a  point  that  can  readily  be  reached. 

The  accompanying  list  of  prime  numbers  and  factors  will 
be  found  useful  in  reducing  and  factoring  fractions. 


PROVIDENCE,   R.  I. 


49 


PRIME     NUMBERS    AND     FACTORS. 
1    TO    1OOO. 


1 

26 

2x13 

51 

3x17 

76 

22xl9 

2 

27 

33 

52 

22xl3 

77 

7x11 

3 

28 

22x7 

53 

78 

2x3x13 

4 

22 

29 

54 

2x33 

79 

5 

30 

2x3x5 

55 

5x11 

80 

24x5 

6 

2x3 

31 

56 

23x7 

81 

34 

7 

32 

25 

57 

3x19 

82 

2x41 

8 

23 

33 

3x11 

58 

2x29 

83 

9 

32 

34 

2x17 

59 

84 

22x3x7 

10 

2x5 

35 

5x7 

60 

22  x  3  x  5 

85 

5x17 

11 

36 

22x32 

61 

86 

2x43 

12 

22x3 

37 

62 

2x31 

87 

3x29 

13 

38 

2x19 

63 

32x7 

88 

23Xll 

14 

2x7 

39 

3x13 

64 

26 

89 

15 

3x5 

40 

23x5 

65 

5x13 

90 

2x32x5 

16 

24 

41 

66 

2x3x11 

91 

7x13 

17 

42 

2x3x7 

67 

92 

22x23 

18 

2x32 

43 

68 

22xl7 

93 

3x31 

19 

44 

22xll 

69 

3x23 

94 

2x47 

20 

22X5 

45 

32x5 

70 

2x5x7 

95 

5x19 

21 

3x7 

46 

2x23 

71 

96 

25x3 

22 

2x11 

47 

72 

23X32 

97 

23 

48 

24x3 

73 

98 

2X72 

24 

23x3 

49 

72 

74 

2x37 

99 

32xll 

25 

52 

50 

2x52 

75 

3x52 

100 

22X52 

BROWN  &  SHARPE   MFG.  CO. 


101 

131 

161 

7x23 

191 

102 

2x3x17 

132 

22x3xll 

162 

2x34 

192 

26x3 

103 

133 

7x19 

163 

193 

104 

23xl3 

134 

2x67 

164 

22X41 

194 

2x97 

105 

3x5x7 

135 

33x5 

165 

3x5x11 

195 

3x5x13 

106 

2x53 

136 

23xl7 

166 

2x83 

196 

22x72 

107 

137 

167 

197 

108 

22x38 

138 

2  x  3  x  23 

168 

23x3x7 

198 

2x32xll 

109 

139 

109 

132 

199 

110 

2X5X11 

140 

22x5x7 

170 

2x5x17 

200 

23x52 

111 

3x37 

141 

3x47 

171 

32xl9 

201 

3x67 

112 

24x7 

142 

2x71 

172 

22x43 

202 

2x101 

113 

143 

11X13 

173 

203 

7x29 

114 

2x3x19 

144 

24x32 

174 

2x3x29 

204 

22x3xl7 

115 

5x23 

145 

5x29 

175 

52x  7 

205 

5x41 

116 

22X29 

146 

2x73 

176 

24xll 

206 

2x103 

117 

32xl3 

147 

3x72 

177 

3x59 

207 

3-'x23 

118 

2x59 

148 

22x37 

178 

2x89 

208 

24xl3 

119 

7x17 

149 

179 

209 

11x19 

120 

23x3x5 

150 

2  x  3  x  52 

180 

22x32x5 

210 

2x3x5x7 

121 

II2 

151 

181 

211 

122 

2x61 

152 

23xl9 

182 

2x7x13 

212 

22X53 

123 

3x41 

153 

32xl7 

183 

3x61 

213 

3x71 

124 

22x31 

154 

2x7x11 

184 

23x.23 

214 

2x107 

125 

53 

155 

5x31 

185 

5x37 

215 

5x43 

126 

2x32x7 

156 

22x3xl3 

186 

2x3x31 

216 

23x33 

127 

157 

187 

11  X17 

217 

7x31 

128 

27 

158 

2x79 

188 

22x47 

218 

2x109 

129 

3x43 

159 

3x53 

189 

33x7 

219 

3x73 

130 

2x5x13 

160 

25X5 

190 

2x5x19 

220 

22x5xll 

PROVIDENCE,   R.  I. 


51 


221 

13x17 

251 

281 

311 

222 

2x3x37 

252 

22x32x7 

282 

2x3x47 

312 

23x3xl3 

223 

253 

11x23 

283 

313 

224 

25x7 

254 

2X127 

284 

22x71 

314 

2x157 

225 

32x52 

255 

3x5x17 

285 

3x5x19 

315 

32x5x7 

226 

2x113 

256 

28 

286 

2x11  x!3 

316 

22x79 

227 

257 

287 

7x41 

317 

228 

22x3xl9 

258 

2  X  3  x  43 

288 

25X32 

318 

2x3x53 

229 

259 

7x37 

289 

172 

319 

11X29 

230 

2  x  5  X  23 

260 

22x5xl3 

290 

2x5x29 

320 

26x5 

231 

3X7X11 

261 

32x29 

291 

3x97 

321 

3x107 

232 

23x29 

262 

2x131 

292 

22x73 

322 

2x7x23 

233 

263 

293 

323 

17x19 

234 

2x32xl3 

264 

23x3xll 

294 

2x3x72 

324 

22x34 

235 

5x47 

265 

5x53 

295 

5x59 

325 

52X13 

236 

22x59 

266 

2x7x19 

296 

23x37 

326 

2x163 

237 

Bx79 

267 

3x89 

297 

33xll 

327 

3x109 

238 

2x7x17 

268 

22x67 

298 

2x149 

328 

23x41 

239 

269 

299 

13x23 

329 

7x47 

240 

24x3xo 

270 

2x33x5 

300 

22x3x52 

330 

2X3X5X11 

241 

271 

301 

7x43 

331 

242 

2xll2 

272 

24xl7 

302 

2x151 

332 

22X83 

243 

^ 

273 

3x7x13 

303 

3x101 

333 

32X37 

244 

22X61 

274 

2x137 

304 

24X19 

334 

2X167 

245 

5X72 

275 

52Xll 

305 

5x61 

335 

5x67 

246 

2x3x41 

276 

22x3x23 

306 

2x32xl7 

336 

24x3x7 

247 

13x19 

277 

307 

337 

248 

23X31 

278 

2x139 

308 

22x7xll 

338 

2X132 

249 

3x83 

279 

32X31 

309 

3x103 

339 

3x113 

250 

2x53 

280 

23  x  5  x  7 

310 

2x5x31 

340 

22xoxl7 

BROWN  &   SHARPK   MFG.   CO. 


341 

11x31 

371 

7x53 

401 

431 

342 

2x32xl9 

372 

22x3x31 

402 

2x3x67 

432 

24x33 

343 

73 

373 

403 

13x31 

433 

344 

23X43 

374 

2X11X17 

404 

22xl01 

434 

2x7x31 

345 

3x5x23 

375 

3X53 

405 

34X5 

435 

3x5x29 

346 

2x173 

376 

23X47 

406 

2x7x29 

436 

22xl09 

347 

377 

13x29 

407 

11x37 

437 

19x23 

348 

22x3x29 

378 

2x33x7 

408 

23x3xl7 

438 

2  X  3  x  73 

349 

879 

409 

439 

350 

2x52x7 

380 

22xoxl9 

410 

2x5x41 

440 

23xoxll 

351 

33X13 

381 

3x127 

411 

3x137 

441 

32x72 

352 

25Xll 

382 

2X191 

412 

22X103 

412 

2x13x17 

353 

383 

413 

7x59 

443 

354 

2x3x59 

384 

27x3 

414 

2x32x23 

444 

22x3x37 

355 

5x71 

385 

5x7  X  11 

415 

5x83 

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5x89 

356 

22x89 

386 

2x193 

416 

23Xl3 

446 

2x223 

357 

3x7x17 

387 

32X43 

417 

3x139 

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358 

2x179 

388 

22X97 

418 

2x11x19 

448 

26x7 

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389 

419 

449 

360 

23X32X5 

390 

2X3X5X13 

420 

22X3X5X7 

450 

2x32x52 

361 

192 

391 

17x23 

421 

451 

11X41 

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2x181 

392 

23X72 

422 

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22X113 

363 

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393 

3x131 

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32X47 

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364 

22x7xl3 

394 

2x197 

424 

23x53 

454 

2x227 

365 

5x73 

395 

5x79 

425 

52xl7 

455 

5x7x13 

366 

2x3x61 

396 

22X32XH 

426 

2x3x71 

456 

23x3xl9 

367 

397 

427 

7x61 

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368 

24X23 

398 

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428 

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458 

2x229 

369 

32x41 

399 

3x7x19 

429 

3x11x13 

459 

-33xl7 

370 

2x5x37 

400 

24x52 

430 

2x5x43 

460 

22x5x23 

PROVIDENCE,    R.  I. 


53 


461 

491 

521 

551 

19x29 

462 

2X3X7X11 

492 

22x3x41 

,522 

2x32x29 

552 

23x3x23 

463 

493 

17x29 

523 

553 

7x79 

464 

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494 

2x13x19 

524 

22xl31 

554 

2x277 

465 

3x5x31 

495 

32x5xll 

525 

3x52x7 

555 

3x5x37 

466 

2  X  233 

496 

24X31 

526 

2x263 

556 

22xl39 

467 

497 

7x  71 

527 

17x31 

557 

468 

22x32xl3 

498 

2x3x83 

528 

24x3xll 

558 

2x32x31 

469 

7x67 

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529 

232 

559 

13x43 

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2x5x47 

500 

22x53 

530 

2x5x53 

560 

24x5x7 

471 

3x157 

501 

3x167 

531 

32X59 

561 

3x11x17 

472 

23x59 

502 

2x251 

532 

22x7xl9 

562 

2x281 

473 

11X43 

503 

533 

13X41 

563 

474 

2x3x79 

504 

23X32X7 

534 

2x3x89 

564 

22x3x47 

475 

52X19 

505 

5x101 

535 

5x107 

565 

5x113 

476 

22x7xl7 

506 

2  X  1  1  X  23 

536 

2sx67 

566 

2x283 

477 

32x53 

507 

3xl32 

537 

3x179 

567 

34x7 

478 

2x239 

508 

22xl27 

538 

2x269 

568 

23X71 

479 

509 

539 

72xll 

569 

480 

25  x  3  x  5 

510 

2X3X5X17 

540 

22x33x5 

570 

2x3X5X19 

481 

13x37 

511 

7x73 

541 

571 

482 

2x241 

512 

29 

542 

2x271 

572 

22xllXl3 

483 

3  X  7  X  23 

513 

33xl9 

543 

3x181 

573 

3x191 

484 

22xll2 

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2  X  257 

544 

25xl7 

574 

2x7x41 

485 

5x97 

515 

5x103 

545 

5x109 

575 

52x23 

486 

2x35 

516 

22x3x43 

546 

2X3X7X13 

576 

26x32 

487 

517 

11X47 

547 

577 

488 

23x61 

518 

2x7x37 

548 

22xl37 

578 

2X172 

489 

3x163 

519 

3x173 

549 

32x61 

579 

3x193 

490 

2  X  5  X  72 

520 

23x5xl3 

550 

2x52xll 

580 

22x5x29 

54 


BROWN  &  SHARPE  MFG.   CO. 


581 

7x83 

611 

13x47 

641 

671 

11X61 

582 

2x3x97 

612 

22x32xl7 

642 

2x3x107 

672 

25  x  3  x  7 

583 

11x53 

613 

643 

673 

584 

23x73 

614 

2x307 

644 

22x7x23 

674 

2x337 

585 

32xoxl3 

615 

3x5x41 

645 

3x5x43 

675 

33x52 

586 

2x293 

616 

23x7xll 

646 

2x17x19 

676 

22xl32 

587 

617 

647 

677 

588 

22x3x72 

618 

2x3x103 

648 

23x34 

678 

2x3x113 

589 

19x31 

619 

649 

11X59 

679 

7x97 

590 

2x5x59 

620 

22x5x31 

650 

2x52xl3 

680 

23x5xl7 

591 

3x197 

621 

33x23 

651 

3x7x31 

681 

3x227 

592 

24x37 

622 

2x311 

652 

22xl63 

682 

2x11x31 

593 

623 

7x89 

653 

683 

594 

2x33Xll 

624 

24x3xl3 

654 

2x3x109 

684 

22x32xl9 

595 

5x7x  17 

625 

54 

655 

5x131 

685 

5x137 

596 

22X149 

626 

2x313 

656 

24x41 

686 

2x73 

597 

3x199 

627 

3x11x19 

657 

32x73 

687 

3x229 

598 

2x13x23 

628 

22xl57 

658 

2x7x47 

688 

24x43 

599 

629 

17x37 

659 

689 

600 

23x3xo2 

630 

2X32X5X7 

660 

22X3X5xll 

690 

2X3x5X23 

601 

631 

661 

691 

602 

2x7x43 

632 

23x79 

662 

2x331 

692 

22X173 

603 

32x67 

633 

3x211 

663 

3x13x17 

693 

32x7xll 

604 

22X151 

634 

2x317 

664 

23x83 

694 

2x347 

605 

5xll2 

635 

5x127 

665 

5x7x19 

695 

5x139 

606 

2x3x101 

636 

22x3x53 

666 

2x32x37 

696 

23x3x29 

607 

637 

72X13 

667 

23x29 

697 

17x41 

608 

25xl9 

638 

2x11x29 

668 

22xl67 

698 

2x349 

609 

3  X  7  X  29 

639 

32X71 

669 

3x223 

699 

3x233 

610 

2x5x61 

640 

27x5 

670 

2x5x67 

700 

22x52x7 

PROVIDENCE,   R.  I. 


55 


701 

731 

17x43 

761 

791 

7x113 

702 

2x33xl3 

732 

22x3x61 

762 

2x3x127 

792 

23x32xll 

703 

19x37 

733 

763 

7x109 

793 

13x61 

704 

2°xll 

734 

2  X  367 

764 

22xl91 

794 

2x397 

705 

3x5x47 

735 

3  X  5  x  72 

765 

32x5xl7 

795 

3x5x53 

706 

2x353 

736 

25x23 

766 

2  x  383 

796 

22X199 

707 

7x101 

737 

11X67 

767 

13x59 

797 

708 

22x3x59 

738 

2x32x41 

768 

28x3 

798 

2X3X7X19 

709 

739 

769 

799 

17x47 

710 

2x5x71 

740 

22x5x37 

770 

2X5X7X11 

800 

25x52 

711 

32X79 

741 

3x13x19 

771 

3x257 

801 

32x89 

712 

23x89 

742 

2x7x53 

772 

22xl93 

802 

2x401 

713 

23x31 

743 

773 

803 

11  X73 

714 

2X3X7X17 

744 

2"x3x31 

774 

2x32x43 

804 

22x3x67 

715 

5x11x13 

745 

5x149 

775 

52x31 

805 

5  x  7  x  23 

716 

22X179 

746 

2x373 

776 

23x97 

806 

2x13x31 

717 

3  X  239 

747 

32x83 

777 

3x7x37 

807 

3  x  269 

718 

2x359 

748 

22xllxl7 

778 

2x389 

808 

23X101 

719 

749 

7x107 

779 

19x41 

809 

720 

24x32x5 

750 

2  x  3  x  53 

780 

22X3X5X13 

810 

2x34x5 

721 

7x103 

751 

781 

11x71 

811 

722 

2xl92 

752 

24x47 

782 

2  x  17x23 

812 

22x7x29 

723 

3x241 

753 

3x251 

783 

33x29 

813 

3x271 

724 

22xl8l 

754 

2x13x29 

784 

24x72 

814 

2x11x37 

725 

52X29 

755 

5x151 

785 

5x  ).">7 

815 

5x163 

726 

2x3xll2 

756 

22x33x7 

786 

2x3x131 

816 

24x3xl7 

727 

757 

787 

817 

19x43 

728 

23x7xl3 

758 

2x379 

788 

22xl97 

818 

2x409 

729 

36 

759 

3x11x23 

789 

3x263 

819 

32X7X13 

730 

2x5x73 

700 

23x5xl9 

790 

2x5x79 

820 

22x5x41 

BROWN  &  SHARPE  MFG.   CO. 


821 

851 

23x37 

881 

911 

822 

2x3x137 

852 

22x3x71 

882 

2x32x72 

912 

24x3xl9 

823 

853 

883 

913 

11x83 

824 

23xl03 

854 

2x7x61 

884 

22xl3xl7 

914 

2x457 

825 

3x52xll 

855 

32x5xl9 

885 

3x5x59 

915 

3x5x61 

826 

2x7x59 

856 

23xl07 

886 

2x443 

916 

22x229 

827 

857 

887 

917 

7x131 

828 

22x32x23 

858 

2x3x11x13 

888 

23x3x37 

918 

2x33xl7 

829 

859 

889 

7x127 

919 

830 

2  X  5  x  83 

860 

22x5x43 

890 

2  x  5  X  89 

920 

23x5x23 

831 

3x277 

861 

3x7x41 

891 

34xll 

921 

3x307 

832 

26xl3 

862 

2x431 

892 

22x223 

922 

2x461 

833 

72xl7 

863 

893 

19x47 

923 

13x71 

834 

2x3x  139 

864 

2s  X33 

894 

2x3x149 

924 

22x3X7XH 

835 

5x167 

865 

5x173 

895 

5x179 

925 

52X37 

836 

22xllXl9 

866 

2x433 

896 

27x7 

926 

2x463 

837 

33x31 

867 

3xl72 

897 

3x13x23 

927 

32xl03 

838 

2x419 

868 

22x7x31 

898 

2x449 

928 

25x29 

839 

869 

11X79 

899 

29x31 

929 

840 

23X3X5X7 

870 

2X3X5X29 

900 

22x32x52 

930 

2X3X5X31 

841 

292 

871 

13x67 

901 

17x53 

931 

72X19 

842 

2x421 

872 

23X109 

902 

2x11x41 

932 

22x233 

843 

3x281 

873 

32x97 

903 

3x7x43 

933 

3x311 

844 

22x211 

874 

2x19x23 

904 

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934 

2x467 

845 

5X132 

875 

53X7 

905 

5x181 

935 

5xllXl7 

846 

2x32x47 

876 

22x3x73 

906 

2x3x151 

936 

23x32X  13 

847 

7xll2 

877 

907 

937 

848 

24X53 

878 

2x439 

908 

22X227 

938 

2x7x67 

849 

3x283 

879 

3  x  293 

909 

32xlOl 

939 

3x313 

850 

2x52xl7 

880 

24xoxll 

910 

2X5X7X13 

940 

22x5x47 

PROVIDENCE,   R.   I. 


57 


941 

956 

22x239 

971 

986 

2x17x29 

942 

2x3x157 

957 

3x11x29 

972 

22x35 

987 

3x7x47 

943 

23x41 

958 

2x479 

973 

7x139 

988 

22x  13x19 

944 

24X59 

959 

7x137 

974 

2x487 

989 

23x43 

945 

33xox7 

960 

2°x3xo 

975 

3x52xl3 

990 

2x32X5Xll 

946 

2x11  X43 

961 

312 

976 

24x61 

991 

947 

962 

2x13x37 

977 

992 

25x31 

948 

22x3x79 

963 

32xl07 

978 

2x3x163 

993 

3x331 

949 

13x73 

964 

22x241 

979 

11X89 

994 

2x7x71 

950 

2x52xl9 

965 

5x193 

980 

22x5x72 

995 

5x199 

951 

3x317 

966 

2X3X7X23 

981 

32xl09 

996 

22x3x83 

952 

23x7xl7 

967 

982 

2x491 

997 

953 

968 

23xll2 

983 

998 

2x499 

954 

2x32x53 

969 

3x17x19 

984 

23x3x41 

999 

33x37 

955 

5x191 

970 

2x5x97 

985 

5x197 

1000 

23x53 

58  BROWN  &   SHARPE  MFG.   CO. 


CHAPTER    VI. 

INTERNAL  GEARING. 

PART  A.— INTERNAL  SPUR  GEARING. 
(Figs.  12,  13,  14,  15,  16.) 

A  little  consideration  will  show  that  a  tooth  of  an  internal 
or  annular  gear  is  the  same  as  the  space  of  a  spur — external 
gear. 

We  prefer  the  epicycloidal  form  of  tooth  in  this  class  of 
gearing  to  the  involute  form,  for  the  reason  that  the  difficulties 
in  overcoming  the  interference  of  gear  teeth  in  the  involute 
system  are  considerable.  Special  constructions  are  required 
when  the  difference  between  the  number  of  teeth  in  gear  and 
pinion  is  small. 

In  using  the  system  of  epicycloidal  form  of  tooth  in  which 
the  gear  of  15  teeth  has  radial  flanks,  this  difference  must  be 
at  least  15  teeth,  if  the  teeth  have  both  faces  and  flanks.  Gears 
fulfilling  this  condition  present  no  difficulties.  Their  pitch 
diameters  are  found  as  in  regular  spur  gears,  and  the  inside 
diameter  is  equal  to  the  pitch  diameter,  less  twice  the  adden- 
dum. 

If,  however,  this  difference  is  less  than  15,  say  6,  or  2,  or  i, 
then  we  may  construct  the  tooth  outline  (based  on  the  epicy- 
cloidal system)  in  two  different  ways. 

First  Method. — To  explain  this  method  better,  let  us  sup- 
pose the  case  as  in  Fig.  T2,  in  which  the  difference  between 
gear  and  pinion  is  more  than  15  teeth.  Here  the  point  o  of 
the  describing  circle  B  (the  diameter  of  which  in  the  best 
practice  of  the  present  day  is  equal  to  the  pitch  radius  of  a  15 
tooth  gear,  of  the  same  pitch  as  the  gears  in  question)  gene- 
rates the  cycloid  o,  o1,  o2,  o3,  etc.,  when  rolling  on  pitch  circle 
L  L  of  gear,  forming  the  face  of  tooth  ;  and  when  rolling  on 
the  outside  of  L  L  the  flank  of  the  tooth.  In  like  manner  is  the 
face  and  flank  of  the  pinion  tooth  produced  by  B  rolling  out- 
side and  inside  of  E  E  (pitch  circle  of  pinion).  A  little  study 


PROVIDENCE,   R.  I. 


59 


of  Fig.    12    (in   which  the  face  and   flank  of  a  gear  tooth  are 
produced)  will   show  the  describing  circle  B  divided  into  12 


equal  parts  and  circles  laid  through  these  points  (i,  2,  3,  etc.), 
concentric  with  L  L.  We  now  lay  off  on  L  L  the  distances 
o— i,  1-2,  2-3,  etc.,  of  the  circumference  of  B,  and  obtain  points 


6o 


BROWN  &  SHARPK   MFG.   CO. 


jl>  2\  3\  etc.  [Ordinarily  it  is  sufficient  to  use  the  chord.]  It 
will  now  readily  be  seen  that  B  in  rolling  on  L  L  will  success- 
ively come  in  contact  with  i1,  2',  3',  etc.,  c  meanwhile  moving 
to  c\  <ra,  <r3,  etc.  (points  on  radii  through  i1,  2',  3*,  etc.),  and  the 
generating  point  o  advancing  to  o1,  o2,  o3,  etc.,  being  the  inter- 
sections of  B  with  flt  <r2,  <r*,  etc.,  as  centers  and  the  circles  laid 
through  i,  2,  3,  etc.  Points  o,  o1,  oa,  o3,  etc.,  connected  with  a 
curve  give  the  face  of  the  tooth  ;  in  like  manner  the  flank  is 
obtained. 

In  this  manner  the  form  of  tooth  is  obtained,  when  the 
difference  of  teeth  in  gear  and  pinion  is  less  than  15,  with  the 
exception  that  the  diameter  of  describing  circle  B 


where  P  =  diametral  pitch,  N  and  n  number  of  teeth  in  gears. 
The  distances  of  the  tooth  above  and  below  the  pitch  line 
as  well  as  the  thickness  /  are  determined  as  in  regular  spur 
gears  by  the  pitch,  except  when  the  difference  in  gear  and 
pinion  is  very  small,  where  we  obtain  a  short  tooth,  as  in  Figs. 
13  and  14.  In  such  a  case  the  height  of  tooth  is  arbitrary  and 
only  conditioned  by  the  curve.  In  internal  gears  it  is  best  to 
allow  more  clearance  at  bottom  of  tooth  than  in  ordinary  spur 
gears. 


42  T. 


8  P. 


30  Teeth 


Fig.  13. 


In  a  construction  of  this  kind  it  is  suggested  to  draw  the 
tooth  outline  many  times  full  size  and  reduce  by  photography. 
An  equally  multiplied  line  A  B  will  help  in  reducing. 


PROVIDENCE,   R.  I. 


61 


62  BROWN  &  SHARPK   MFG.   CO. 

Second  Method. — The  difference  between  gear  and  pinion 
being  very  small,  it  is  sometimes  desirable  to  obtain  a  smooth 
action  by  avoiding  what  is  termed  the  "  friction  of  approach- 
ing action."*  This  is  done,  the  pinion  driving,  by  giving  gear 
only  flanks,  Fig.  15,  and  the  gear  driving,  by  giving  gear  only 
faces,  Fig.  16.  In  both  these  cases  we  have  but  one  describ- 
ing circle,  whose  diameter  is  equal  to  the  difference  of  the  two 
pitch  diameters.  The  construction  of  the  curve  is  precisely 
the  same  as  described  under  A.  The  describing  circle  has 
been  divided  into  24  parts  simply  for  the  sake  of  greater 
accuracy. 


PART  B.—  INTERNAL  BEVEL  GEARS. 


The  pitch  surfaces  of  bevel  gears  are  cones  whose  apexes 
are  at  a  common  point,  rolling  upon  each  other.  The  tooth 
forms  for  any  given  pair  of  bevel  gears  are  the  same  as  for  a 
pair  of  spur  gears  (of  same  pitch)  whose  pitch  radii  are  equal  to 
the  respective  apex  distances  of  the  normal  cones  (/.  <?.,  cones 
whose  elements  are  perpendicular  upon  the  elements  of  the 
bevel  gear  pitch  cones).  (Compare  Fig  19,  page  68-) 

The  same  is  true  of  internal  bevel  gears,  with  the  modifica- 
tion that  here  one  of  the  pitch  cones  rolls  inside  of  the  other. 
The  spur  gears  to  whose  tooth  forms  the  forms  of  the  bevel 
gear  teeth  correspond,  resolve  themselves  into  internal  spur 
gears  (Fig.  17).  The  problem  is  now  to  be  solved  as  indicated 
in  the  first  part  of  this  chapter. 


McCord,  Kinematics,  pages  107,  108. 


PROVIDENCE,   R.   I. 


8  P. 

Gear    4O  Teeth 
Pinion    20  Teeth 


Fig. 


64  BROWN  &  SHARPS   MFG.   CO. 


CHAPTER  vn. 

V 

GEAR    PATTERNS. 

(Fig.  18.) 

To  place  in  bevel  gears  the  best  iron  where  it  belongs,  the 
tooth  side  of  the  pattern  should  always  be  in  the  nowel,  no 
matter  of  what  shape  the  hubs  are. 

Hubs,  if  short,  may  be  left  solid  on  web  ;  if  long  they  should 
be  made  loose.  A  long  hub  should  go  on  a  tapering  arbor,  to 
prevent  tipping  in  the  sand.  i°  taper  for  draft  on  hubs  when 
loose,  and  3°  when  solid  is  considered  sufficient. 

Coreprints  as  a  rule  are  made  separate,  partly  to  allow  the 
pattern  to  be  turned  on  an  arbor,  partly  for  convenience, 
should  it  be  desirable  to  use  different  sizes. 

Put  rap-  and  draw-holes  as  near  to  center  as  possible. 
Referring  to  Fig.  18,  make  L  —  D  for  D  from  %"  to  i/^",  or 
even  more,  should  hubs  be  very  long.  Otherwise  if  D  is  more 
than  1%"  leave  L  =  i%". 

Iron  pattern  before  using  should  be  marked,  rusted  and 
waxed. 

Shrinkage — For  cast-iron,  ^6"  per  foot. 
For  brass,        -f$"        " 

Cast-iron  gears,  especially  arm  gears,  do  not  always  shrink 
Hff  per  foot.  In  making  iron  patterns  the  following  allow- 
ances have  been  found  useful : 

Up  to  12"  diameter  allow  no  shrink. 
From  12"  to  18"         "  "      ^  regular  shrink. 

"         I8"t024"  «  "         %          " 

"      24"  to  48"         "  "      ft       " 

Above  48"        "  "     .10"    " 

for  cast-iron. 


PROVIDENCE,    R.    I. 


66 


BROWN  &  SHARPK  MFG.   CO. 


If  in  gears  the  teeth  are  to  be  cast,  the  tooth  thickness  t  in 
the  pattern  is  made  smaller  than  called  for  by  the  pitch,  to  avoid 
binding  of  the  teeth  when  cast.  No  definite  rule  can  be  given, 
as  the  practice  varies  on  this  point.  For  the  different  diam- 
etral pitches  we  would  advise  making  /  smaller  by  an  amount 
expressed  in  inches,  as  given  in  the  following  table  : 


DIAM.  PITCH. 

AMOUNT  t 
is  SMALLER. 

DIAM.  PITCH. 

AMOUNT  t 
is  SMALLER. 

16 

.010" 

5 

.020" 

12 

.012" 

4 

.022" 

10 

.014" 

3 

.026" 

8 

.016" 

2 

.030" 

6 

.018" 

I 

.040" 

PROVIDENCE,    R.  I.  67 


vui. 

DIMENSIONS   AND    FORM    FOR    BEVEL    GEAR 

CUTTERS. 

(Fig.   19.) 

The  data  needed  to  determine  the  form  and  thickness  of  a 
bevel  gear  cutter  are  the  following  : 
P  =  pitch. 

N  =  number  of  teeth  in  large  gear. 
n=  number  of  teeth  in  small  gear. 
F  =  length  of  face  of  tooth,  measured  on  pitch  line. 
After  having  laid  out  a  diagram  of  the  pitch  cones  a  b  c  and 
a  b  f,  and  laid  off  the  width  of  face,  the  problem  resolves  itself 
into  two  parts  : 

PART  I.— DETERMINE  PROPER  CURVE  FOR  CUTTER. 
It  will  be  remembered  that  in  the  involute  system  of  cutters 
(the  only  one  used  for  bevel  gears  that  are  cut  with  rotary 
cutter),  a  set  of  eight  different  cutters  is  made  for  each 
pitch,  numbering  from  No.  i  to  No.  8,  and  cutting  from 
a  rack  to  12  teeth.  Each  number  represents  the  form  of 
a  cutter  suitable  to  cut  the  indicated  number  of  teeth.  For 
instance,  No.  4  cutter  (No.  4  curve)  will  cut  26  to  34  teeth. 
In  order  to  find  the  curve  to  be  used  for  gear  and  pinion 
we  simply  construct  the  normal  pitch  cones  by  erecting 
the  perpendicular  p  q  through  b,  Fig.  19.  We  now  measure  the 
lines  b  q  and  b  p,  and  taking  them  as  radii,  multiplying  each  by 
2  and  P  we  obtain  a  number  of  teeth  for  which  cutters  of 
proper  curves  may  be  selected.  From  example  we  have  : 

Gear :     b  q  -  9^"  ;   2  X  P  X  9.75  =  97  T       No.  2  curve. 

Pinion:  bp  =  3^"  ;  2  X  P  X  3-5    =  35  T      No-  3  curve. 
The  eight  cutters  which  are  made  in  the  involute  system 
for  each  pitch  are  as  follows  : 

No.  i  will  cut  wheels  from  135  teeth  to  a  rack. 

2  "  "          "        55     "       "  134  teeth. 

3  "  "  "        35     "       "     54     " 

4  «  «  "        26     "        "     34     " 

c-  "  "  '*  21       "  "       2<        " 

6  "  "  "        17     "        "     20      " 

7  «  <<  «  I4      «          «i       I(j        << 

8  «  "  «        12     "        "     13      " 


68 


BROWN  &  SHARPE   MFG.  CO. 


PROVIDENCE,    R.   I.  69 

PART  II.  — DETERMINE   THICKNESS  OF  CUTTER. 

It  is  very  evident  that  a  bevel  gear  cutter  cannot  be  thicker 
than  the  width  of  the  space  at  small  end  of  tooth  ;  the  practice 
is  to  make  cutter  .005"  thinner.  Theoretically  the  cutting  angle 
(h}  is  equal  to  pitch  angle  less  angle  of  bottom  (or  h  =  a  —  ft'). 
Practically,  however,  better  results  are  obtained  by  making 
h  =  a  —  ft  (substituting  angle  of  top  for  angle  of  bottom),  and 
in  calculating  the  depth  at  small  end,  to  add  the  full  clearance 
(/)  to  the  obtained  working  depth,  giving  equal  amount  of 
clearance  at  large  and  small  end.  This  is  done  to  obtain  a 
tooth  thinner  at  the  top  and  more  curved.  As  the  small  end 
of  tooth  determines  the  thickness  of  cutter,  we  shall  have  to 
find  the  tooth  part  values  at  small  end.  From  the  diagram  it 
will  be  seen  that  the  values  at  large  end  are  to  those  at  small 
end  as  their  respective  apex  distances  (a  b  and  a  I).  The 
numerical  values  of  these  can  be  taken  from  the  diagram  and 
the  quotient  of  the  larger  in  the  smaller  is  the  constant  where- 
with to  multiply  the  tooth  values  at  large  end,  to  obtain  those 
at  small  end.  In  our  example  we  find  : 

a  7  =  —  =  .655  =  constant  T? 

a  b  =    .8  For  5  P  we  have  : 


^  =  .2057       . 

S  =  .2000  /  =  .I3IO 

/=-03I4  /=.Q3i4 

^+/=.23i4  /+/=.i624 

D"  +  /  =  .4314.  J'  =  .1310 

D'"+/=.2934 

From  the  foregoing  it  is  evident  that  a  spur  gear  cutter 
could  not  be  used,  since  a  bevel  gear  cutter  must  be  thinner, 

If  in  gears  of  more  than  30  teeth  the  faces  are  proportion- 
ately long,  we  select  a  cutter  whose  curve  corresponds  to  the 
midway  section  of  the  tooth.  The  curve  of  the  cutter  is  found 
by  the  method  explained  in  Part  I.  of  this  Chapter. 


BROWN  &   SHARPK   MFG.  CO. 


IX. 


DIRECTIONS   FOR  CUTTING   BEVEL    GEARS 
WITH  ROTARY  CUTTER. 

(Fig.     20.) 

In  order  to  obtain  good  results,  the  gear  blanks  must  be  of 
the  right  size  and  form.     The  following  sizes  for  each  end  of 
the  tooth  must  be  given  the  workman  : 
Total  depth  of  tooth. 
Thickness  of  tooth  at  pitch  line. 
Height  of  tooth  above  pitch  line. 

These  sizes  are  obtained  as  explained  in  Chapter  VIII. 
The   workman  must  further  know  the  cutting  angle  (see 
formula  on  page  13  and  compare  Chapter  VIII.),  and  be  pro- 
vided with  the  proper  tools  with  which  to  measure  teeth,  etc. 
In  cutting  a  gear  on  a  universal  milling  machine  the  opera- 
tions and  adjustments  of  the  machine  are  as  follows  : 

1.  Set  spiral  bed  to  zero  line. 

2.  Set  cutter  central  with  spiral  head  spindle. 

3.  Set  spiral  head  to  the  proper  cutting  angle. 

4.  Set  the  index  on  head  for  the  number  of  teeth  to  be  cut, 
leaving  the  sector  on  the  straight  or  numbered  row  of  holes, 
and  set  the  pointer  (or  in  some  machines  the  dial)  on  cross-feed 
screw  of  milling  machine  to  zero  line. 

5.  As  a  matter  of  precaution,  mark  the  depth  to  be  cut  for 
large  and  small  end  of  tooth  on  their  respective  places. 

6.  Cut  two  or  three   teeth  in  blank  to  conform  with  these 
marks  in  depth.     The  teeth  will  now  be  too  thick  on  both  their 
pitch  circles. 

7.  Set  the  cutter  off  the  center  by  moving  the  saddle  to  or 
from  the  frame  of  the  machine  by  means  of   the  cross-feed 
screw,  measuring  the  advance  on   dial  of  same.     The  saddle 
must   not   be    moved    further  than  what   to   good   judgment 


PROVIDENCE,    R.   I. 


Fig.  20. 


72  BROWN  &   SHARPE  MFG.  CO. 

appears  as  not  excessive  ;  at  the  same  time  bearing  in  mind 
that  an  equal  amount  of  stock  is  to  be  taken  off  each  side  of 
tooth. 

8.  Rotate  the  gear  in  the  opposite  direction  from  which  the 
saddle  is  moved  off  the  center,  and  trim  the  sides  of  teeth  (A) 
(Fig.  20'.) 

9.  Then  move  the  saddle  the  same  distance  on  the  opposite 
side  of  center  and  rotate  the  gear  an  equal  amount  in  the 
opposite  direction  and  trim  the  other  sides  of  teeth  (C). 

10.  If  the  teeth  are  still  too  thick  at  large  end  E,  move  the 
saddle  further  off  the  center  and  repeat  the  operation,  bearing 
in  mind  that  the  gear  must  be  rotated  and  the  saddle  moved 
an  equal  amount  each  way  from  their  respective  zero  settings. 

It  is  generally  necessary  to  file  the  sides  of  teeth  above  the 
pitch  line  more  or  less  on  the  small  ends  of  teeth,  as  indicated 
by  dotted  lines  F  F.  This  applies  to  pinions  of  less  than  30 
teeth. 

For  gears  of  coarser  pitch  than  5  diametral  it  is  best  to 
make  one  cut  around  before  attempting  to  obtain  the  tooth 
thickness. 

The  formulas  for  obtaining  the  dimensions  and  angles  of 
gear  blanks  are  given  in  Chapter  III. 


PROVIDENCE,    R.   I. 


73 


THE    INDEXING   OF  ANY  WHOLE   OR    FRAC- 
TIONAL  NUMBER. 

(Fig.    21.) 


•Change  Gear 
Fig.  21. 


In  indexing  on  a  machine  the  question  simply  is  :  How 
many  divisions  of  the  machine  index  have  to  be  advanced  to 
advance  a  unit  division  of  the  number  required.  To  which 
is  the 

divisions  of  machine  index 
answer  =  — 

number  to  be  indexed 

Suppose  the  number  of  divisions  in  index  wheel  of  machine 
to  be  216. 


EXAMPLE  I.  —  Index  72. 


Answer: 


216 
-  =  3 


/ 

(3  turns  of  worm). 


74  BROWN  &  SHARPS  MFG.   CO. 

EXAMPLE  II. — Index  123. 

—  =  i  +  .93 
123  123 

If  now  we  should  put  on  worm  shaft  a  change  gear  having 
123  teeth,  give  the  worm  shaft,  Fig.  21,  one  turn,  and  in  addi- 
tion thereto  advance  93  teeth  of  the  change  gear  (to  give  the 
fractional  turn),  we  would  have  indexed  correctly  one  unit  of 
the  given  number,  and  so  solved  the  problem.  Should  we  not 
have  change  gear  123  we  may  try  those  on  hand.  The  ques- 
tion then  is  :  How  many  teeth  (x)  of  the  gear  on  hand  (for 
instance  82)  must  we  advance  to  obtain  a  result  equal  to  the 
one  when  advancing  93  teeth  of  the  123  tooth  gear?  We  have  : 

-93.  ^  _X_  where  y  =  62 
123      82 

EXAMPLE  III. — Index  365,  change  gear  147. 

—  ==  -%-  where  j  =  87  —  JL 

365      147  365 

Here  147  is  the  change  gear  on  hand.  In  indexing  for  a  unit 
of  365  we  advance87teeth  of  our  147  tooth  gear.  It  is  evident 
that  in  so  doing  we  advance  too  fast  and  will  have  indexed 
three  teeth  of  our  change  gear  too  many  when  the  circle  is 
completed.  To  avoid  having  this  error  show  in  its  total  amount 
between  the  last  and  the  first  division,  we  can  distribute  the 
error  by  dropping  one  tooth  at  a  time  at  three  even  intervals. 

EXAMPLE  IV. — Index  190. 

216  __  26 

^,  ~          ^90          Change  gear  on  hand  88  T 

26        y      ,  8 

—  =  A  where  j  =  12  + 

190     88  190 

To  distribute  the  error  in  this  case  we  advance  one  additional 
tooth  ot  a  time  of  the  change  gear  at  eight  even  intervals. 

EXAMPLE  V.— Index  117.3913. 

216      _          986087 


"739I3 

This  example  is  in  nowise  different  from  the  preceding 
ones,  except  that  the  fraction  is  expressed  in  large  numbers 
This  fraction  we  can  reduce  to  lower  approximate  values, 
which  for  practical  purposes  are  accurate  enough.  This  is 
done  by  the  method  of  continued  fractions.  [For  an  explana- 


PROVIDENCE,    R.   I.  75 

tion  of  this  method  we  refer  to  our  "  Practical  Treatise  on 
Gearing."] 

986087 
1173913 

986087)  1173913  (i 
986087 

187826)  986087  (5 
939130 

46957)  187826  (3 
140871 

46955)  46957  Ci 
46955 

2)  46955  (23477 
46954 

1)2(2 
2 

986087    =T 
"73913 


i  +  i 


23477  +  1 

2 


<r=3    i   23477 


{r=i   b  =  5   d '=  16  21  493033   986087 
al=i  bl  —  6  dl  =  ig  25  586944  1173913 

NOTE. — Find  the  first  two  fractions  by  reduction       =  -  and  —     -  =  -  ;     the 


others  are  then  found  by  the  rule  \  b  c  +  a  ~  d 

\  t>1  c  -j-  a1  = 


i  +  i      6 
5 


The  fraction  |J-  is  a  good  approximation;  putting  therefore 
a  change  gear  of  25  teeth  on  worm  shaft,  we  advance  (beside 
the  one  full  turn)  21  teeth  to  index  our  unit. 

Of  course,  in  using  any  but  the  correct  fraction  we  have  an 
error  every  time  we  index  a  division  ;  so  that  when  indexed 
around  the  whole  circle,  we  have  multiplied  this  error  by  the 
number  of  divisions. 

In  the  present  example  this  error  is  evidently  equal  to  the 
difference  between  the  correct  and  the  approximate  fraction 
used.  Reducing  both  common  fractions  to  decimal  fractions 
we  have  : 

=  .84000006 


1*739*3 

21 

~ 


.00000006  =  error  in  each  division. 


76  BROWN  &  SHARPE  MFG.   CO. 

.00000006  X  117.3913  —  .00000704348  total  error  in  complete 
circle.  This  error  is  expressed  in  parts  of  a  unit  division.  (To 
find  this  error  expressed  in  inches,  multiply  it  by  the  distance 
between  two  divisions,  measured  on  the  circle.)  In  this  case 
the  approximate  fraction  being  smaller  than  the  correct  one, 
in  indexing  the  whole  circle  we  fall  short  .00000704348  of  a 
division. 

EXAMPLE  VI. — Index  15.708 

216    _  11796 


15.708  15708 

11796  _  983 
15708      1309 
983)  1309  (i 
983 

326)  983  (3 
978 

5)  326  (65 
30 
26 

I5 

D5(S 
5 
o 

983  =I 
'309      T^i__ 

3+x_ 
65  +  1 
5 

i         3          65  5 

i         3         *96          983 
i         4         261          1309 


In  using  the  approximation  £j-J  the  error  for  each  division 
(found  as  above)  will  be  .000002927,  for  the  whole  circle 
.0000460.  In  this  case,  the  approximation  being  larger  than 
the  correct  fraction,  we  overreach  the  circle  by  the  error. 


PROVIDENCE,  R.   I. 


77 


CHAPTER    XI. 

THE  GEARING  OF   LATHES   FOR   SCREW 
CUTTING. 

(Figs.  22,  23.) 

The  problem  of  cutting  a  screw  on  a  lathe  resolves  itself  into 
connecting  the  lathe  spindle  with  the  lead  screw  by  a  train  of 
gears  in  such  a  manner  that  the  carriage  (which  is  actuated  by 


Simple  Gearing. 

Fig.  22. 


78  BROWN  &  SHARPK  MFG.   CO. 

the  lead  screw)  advances  just  one  inch,  or  some  definite  dis- 
tance, while  the  lathe  spindle  makes  a  number  of  revolutions 
equal  to  the  number  of  threads  to  be  cut  per  inch. 

The  lead  screw  has,  with  the  exception  of  a  very  few  cases, 
always  a  single  thread,  and  to  advance  the  carriage  one  inch  it 
therefore  makes  a  number  of  revolutions  equal  to  its  number 


Compound  Gearing. 

Fig.  23. 


of  threads  per  inch.  Should  the  lead  screw  have  double 
thread,  it  will,  to  accomplish  the  same  result,  make  a  number 
of  revolutions  equal  to  half  its  number  of  threads  per  inch.  It 
follows  that  we  must  know  in  the  first  place  the  number  of 
threads  per  inch  on  lead  screw. 


PROVIDENCE,   R.   I.  79 

It  ought  to  be  clearly  understood  that  one  or  more  inter- 
mediate gears,  which  simply  transmit  the  motion  received  from 
one  gear  to  another,  in  no  wise  alter  the  ultimate  ratio  of  a 
train  of  gearing.  An  even  number  of  intermediate  gears 
simply  change  the  direction  of  rotation,  an  odd  number  do  not 
alter  it. 

The  gearing  of  a  lathe  to  solve  a  problem  in  screw  cutting 
can  be  accomplished  by 

A.  Simple  gearing. 

B.  Compound  gearing. 

Referring  to  the  diagrams,  Figs.  22  and  23,  we  have  in  Fig. 
22  a  case  of  simple,  and  in  Fig.  23  a  case  of  compound  gear- 
ing. 

In  simple  gearing  the  motion  from  gear  E  is  transmitted 
either  directly  to  gear  Ron  lead  screw  or  through  the  interme- 
diate F.  In  compound  gearing  the  motion  of  E  is  transmitted 
through  two  gears  (G  and  H)  keyed  together,  revolving  on  the 
same  stud  «,  by  which  we  can  change  the  velocity  ratio  of  the 
motion  while  transmitting  it  from  E  to  R.  With  these  four 
variables  E,  G,  H,  R,  we  are  enabled  to  have  a  wider  range  of 
changes  than  in  simple  gearing. 

B  and  C,  being  intermediate  gears,  are  not  to  be  considered. 
If,  as  is  generally  the  case,  gear  A  equals  gear  D,  we  disregard 
them  both,  simply  remembering  that  gear  E  (being  fast  on 
same  shaft  with  D)  makes  as  many  revolutions  as  the  spindle. 
Sometimes  gear  D  is  twice  as  large  as  gear  A,  then,  still  con- 
sidering gear  E  as  making  as  many  revolutions  as  the  spindle, 
we  deal  with  the  lead  screw  as  having  twice  as  many  threads 
per  inch  as  it  measures. 


SIMPLE  GEARING. 

Let  there  be :    the  number  of  teeth  in  the  different  gears 
expressed  by  their  respective  letters,  as  per  Fig.  22,  and 

s  =  threads  per  inch  to  be  cut, 
L  —  threads  per  inch  on  lead  screw  ;  then 

i.  s  ^R 

L      E 


80  BROWN  &  SHARPE  MFG.   CO. 

If  now  one  of  the  two  gears  E  and  R  is  selected,  the  other 
will  be  : 

R  =  lE        E  =  LR 
L  s 

2.  The  two  gears  may  be  found  by  making 

£  ~^  ^  >  where/  may 'be  any  number. 

3.  The  above  holds  good  when  a  fractional  thread  is  to  be 
cut,  but  if  the  fraction  is  expressed  in  large  numbers,  as,  for 
instance,  s  =  2.833  (2T877W)> we  first  reduce  this  fraction  (yWoO  to 
lower  approximate  values  by  the  process  of  continued  fraction 
(see  pages  73  and  74). 

833)  icoo  (i 

833 


i6S)  167  (i 
165 


2)  l65  (82 

16 


I 

4 

i 

0 

82 

2 

I 

I 

S 

5 
6 

497 

833 
1000 

•L  =  .833  (nearly)  and  s  =  2?- 
o  6 

If  in  this  case  L  =  4,  and  we  select  E  =  48,  then,  since 


COMPOUND  GEARING. 

4.  In  a  lathe  geared  compound  for  cutting  a  screw  the 
product  of  the  drivers  (E  and  H,  Fig.  23)  multiplied  by  the  num- 
ber of  threads  per  inch  to  be  cut  must  equal  the  product  of  the 
driven  (G  and  R)  multiplied  by  the  number  of  threads  on  lead 
screw.  This  is  expressed  by 


.  L, 


PROVIDENCE,    R.   I.  8l 

If  three  of  the  gears  E,  H,  G,  R  have  been  selected,  the 
fourth  one  would  be  either 

..«i        o, 

TT      G  R  L 

H  = or 


G  =  n"J          or 


=  R_G_L   ^_L  /  R.G  \ 
EH  VL.E.H/ 


If  a  fractional  thread  is  to  be  cut,  as  under  "  3,"  we  reduce 
the  fraction  to  lower  approximate  values. 

EXAMPLE.— Gear  for  5.2327  threads  per  inch,  lead  screw  is 
6  threads. 


IOOOO 

2327)  looco  (4 
9308 

692)  2327  (3 
2076 

"251)692(2 
502 

190)  251  (i 
190 

61)  190  (3 
183 

7)  61  (8 
S6 

5)7(i 
5_ 

4 

1)2(2 

a 
o 

43213      8       I       2        2 

L  A  _L  15.  37   3°6   343   ^92   2327 
4  13  30  43  159  1315  J474  4263  loooo 

—  —  .2327  (nearly)  and  5.2327  =  5 15 
43  "43 

Selecting  E  —  43,  H  =  52,  R  =  50,  and 


*  we  have  G  =  43  '  5*  •  5      =  39 


R  .  L  50  .  6 


82  BROWN  &  SHARPK  MFG.   CO. 

5.  The  examples  so  far  given  all  deal  with  single  thread. 
The  pitch  of  a  screw  is  the  distance  from  center  of  one  thread  to 
the  center  of  the  next.  The  lead  of  a  screw  is  the  advance  for 
each  complete  revolution.  In  a  single  thread  screw  the  pitch 
is  equal  to  the  lead,  while  in  a  double  thread  screw  the  pitch 
is  equal  to  one-half  the  lead  ;  in  a  triple  thread  screw  equal  to 
one-third  the  lead,  etc. 

If  we  have  to  gear  a  lathe  for  a  many-threaded  screw 
(double,  triple,  quadruple,  etc.),  we  simply  ascertain  the  lead, 
and  deal  with  the  lead  as  we  would  with  the  pitch  in  a  single 
thread  screw,  *'.  e.,  we  divide  one  inch  by  it,  to  obtain  the  num- 
ber of  threads  for  which  we  have  to  gear  our  lathe. 

EXAMPLE.  —  Gear  for  double  thread  screw,  lead  =  .4654. 
Number  of  threads  per  inch  to  be  geared  for  is  : 

Lead       -4654 

Lead  screw  is  four  threads  per  inch. 

As  in  previous  examples,  we  reduce  the  fraction  .i4%7=itf£fo 
to  lower  approximate  values  by  the  process  of  continued  frac- 
tion. 

From  the  different  values  received  in  the  usual  way  we 
select  : 

^J  =  .1487  (nearly)  and  2.1487  =  2ij-J 

We  have  therefore  : 


=74 
Selecting     -j  G  =  30 


R  _  E  .  H  .  s  _  74  .  4Q  .  Hi  -  -- 
G  .  L  30  .  4 

NOTE.  —  In  using  any  but  the  original  fraction  we  commit  an  error.  This  error 
can  be  found  by  reducing  the  approximate  fraction  used  to  a  decimal  fraction,  and 
comparing  it  with  the  original  fraction.  In  the  above  example  the  original  fraction  is 

.1487    and 
H  =  .  14864 
Error  =  .00006  inch  in  lead. 


In  cutting  a  multiple  screw,  after  having  cut  one 
thread,  the  question  arises  how  to  move  the  thread  tool  the 
correct  amount  for  cutting  the  next  thread. 


PROVIDENCE,   R.   I.  83 

In  cutting  double,  triple,  etc.,  threads,  if  in  simple  or  com- 
pound gearing  the  number  of  teeth  in  gear  E  is  divisible  by 
2,  3,  etc.,  we  so  divide  the  teeth  ;  then  leaving  the  carriage 
at  rest  we  bring  gear  E  out  of  mesh  and  move  it  forward  one 
division,  whereby  the  spindle  will  assume  the  correct  position. 

When  E  is  not  divisible  we  find  how  many  turns  (V)  of 
gear  R  are  made  to  each  full  turn  of  the  spindle.  Dividing 
this  number  by  2  for  double,  by  3  for  triple  thread,  etc.,  we 
advance  R  so  many  turns  and  fractions  of  a  turn,  being  careful 
to  leave  the  spindle  at  rest. 

For  compound  gearing : 

V_E.H 
~G~R 

When  the  gear  D  is  twice  as  large  as  the  gear  A  (as  ex- 
plained in  fifth  paragraph,  page  78.)  the  formula  would  be 

y=      E.  H. 

2  G.  R. 

If  in  simple  gearing  both  E  and  R  are  not  divisible,  one 
remedy  would  be  to  gear  the  lathe  compound  ;  or  the  face- 
plate may  be  accurately  divided  in  two,  three  or  more  slots, 
and  all  that  is  then  necessary  is  to  move  the  dog  from  one  slot 
to  another,  the  carriage  remaining  stationary. 


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